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Goldstein, iarvin E.: Braun, Willis H.Advanced Methods for the Solution of DifferentialEquations.National Aeronautics and Space Administration,Cleveland, Ohio. Levis Research Center.NASA-SP-3167.3

353p.Superintendent of Documents, Government PrintingOffice, Washington, D.C. 20402 (Stock ?o. 3300-00478,$3.70)

MP -$0.75 HC- $17.40 PLUS POSTAGE*College Mathematics; *Sathesatical Applications;Mathematics Education; *TextbooksComplex Numbers; Differential Equations

This in a textbook, originally developed forscientists and engineers, which stresses the actual solutions cfpractical problems. Theorems are precisely stated, but the proofs aregenerally omitted. Sample contents include first-order equations,equations in the complex plane, irregular singular points, andnumerical nethoda. A more recent idkia, the theory of matchedasymptotic expansions, is also included. (Auther/LS)

ced Methods for the Solution of

MAL EQUATIONS4 vconati*Otivt to .1 11,001

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NASA SP-316

NATIONAL.AERONAUTICS'AND.SO:!ACE ADMINISTRATION

NASA SP-316

Advanced Methods for the Solution of

DIFFERENTIAL EQUATIONS

Marvin E.Goidsteki and WM H. Braun

Lewis Research CenterCleveland, Ohio

Scientific and Technical Information Office 1973

NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONWashington, D.C.

For sale by the Superintendent of Documents,U.S. Government Printing Office, Washington, D.C. 20402Price $3.70 Stock Number 3300-00478Library of Congress Catalog Card Number 73-600126

CONTENTS

CHAM PAGE

1 INTRODUCTION 1

2 FIRSTORDER EQUATIONS 353 SYSTEMS OF EQUATIONS 494 ELEMENTARY METHODS FOR SECOND-ORDER EQUATIONS. 735 REVIEW OF COMPLEX VARIABLES... 1116 SOLUTION OF LINEAR SECOND-ORDER DIFFERENTIAL EQUA-

TIONS IN THE COMPLEX PLANE... 159RIEMANN-PAPPERITZ EQUATION AND THE HYPERGEOMETRIC

EQUATION 2018 CONFLUENT HYPERGEOMETRIC EQUATION AND CONFLUENCE

OF SINGULARITII1S IN RIEMANN-PAPPERITZ EQUATION 2339 SOLUTIONS NEAR IRREGULAR SINGULAR POINTS: ASYMPTOTIC

EXPANSIONS 25110 EXPANSIONS IN SMALL AND LARGE PARAMETERS: SINGULAR

PERTURBATIONS.. 28111 NUMERICAL METHODS 319

REFERENCES 347INDEX 351

iii

PREFACE

This book is based on a course presented at the Lewis Research Center.The course was given to engineers and scientists who were interested in in-creasing their knowledge of differential equations. Those results which canactually be used to solve equations are therefore emphasized; and detailedproofs of theorems are, for the most part, omitted. However, the conclusionsof the theorems are stated in a precise manner, and enough references aregiven so that the interested reader can find the steps of the proofs.

In spite of the fact that differential equations is a branch of mathematicswhich is to some extent a collection of loosely related ideas, we have attemptedto impart as much unity to the subject as possible and to point out the connec-tion between the various topics. Although the material is not new, some partsof the presentation are unconventional. Certain modern ideas, such as thetheory of matched asymptotic expansions, which have not worked their wayinto most conventional texts, are also included. In the chapter on numericalmethods, we have attempted to include a discussion of the considerationswhich should be taken into account when applying the methods to variousproblems which arise frequently in practice. Techniques for solving first-orderpartial differential equations are discussed in the chapter on systems of ordi-nary differential equations since it is felt that these topics ::.re easier to under-stand when they are presented simultaneously.

CHAPTER 1

Introduction

In this chapter we shall introduce some preliminary concepts and try togain a certain amount of insight into the behavior of differential equations andtheir solutions. Such questions ab what is meant by the solution of a differentialequation and what is the appropriate number of solutions will be consideredwith some care.

Leibniz, in 1676, was probably the first to introduce the term "differentialequation." However, the study of differential equations had its beginningssomewhat before this time in investigations of physical phenomena. Ever since,developments in the field of differential equations have been closely related tothe physical sciences.

1.1 SOLUTION OF EQUATIONS IN GENERAL

In this section we introduce some concepts which are needed for thesubsequent presentation. First, in order to obtain a geometric interpretationof certain results, we shall have occasion to interpret a set of n variables, sayxi, , X/11 as coordinate axes in an n-dimensional space. A particular setof values, say , . . . , x°, of the n variables are then the coordinates of apoint in this space. When n = 2 and n = 3, these spaces are the very familiartwo- and three-dimensional Euclidean spaces. Hence, as will be seen, thisprocedure allows us to use our geometric intuition about two- and three-dimensional spaces to "picture" results about functions and equations involvingany number of variables.

We shall not need to use many of the mathematical properties of then-dimensional spaces; however, it will be important to have a careful defini-tion of certain special regions in these spaces. Thus, let x?, . . xf: be the


coordinates' of a fixed point in an n-dimensional space, and consider thecollection of all points whose coordinates satisfy the inequalities

< 8, . lxn < 8

for some positive number 8. When n=- 2, these points form the interior of asquare with center at (4, 4) and sides of length 28. When n=3, the pointsform the interior of a cube with center at (4, 4, 4) and sides of length 28.The reason for using the word "interior" is that the boundary points whichsatisfy the equalities

(xi x°( =8, . . Ixn-41=8

have been omitted from the collection.Next we define a domain D to be any region of the n-dimensional space

which satisfies the following two conditions: (1) there exists a positive number8 for each point .4 . . 4, belonging to D such that every point xl, . . xnlying in the range

41 < 8, . . Ixn < 8

is also a point of D; and (2) any two points in this region can be connected by acontinuous line. Thus, in a two-dimensional space the first condition requiresthat we be able to draw around each point of D a square whose interior liesentirely within D. This is illustrated in figure 1-1. The only points where thisis not possible are the points which lie on the boundary r of D. This is becauseevery square about a point of must include at least some points which do notbelong to D. Thus, the first condition serves to exclude the boundary pointsfrom D.

The second condition simply requires that the region not be composed oftwo or more disconnected subregions, as shown for two dimensions in figure1-2.

A neighborhood of the point 4, . . x,°, is simply a domain which containsthis point. Evidently, the collection of all points which satisfy the inequalitiesin condition 1 form a neighborhood of the point 4 . .

'Instead of saying that x7. . . . 4 are the coordinates of the point, we shall frequently say that the collection4, . .,1,1is the point.

2

INTRODUCTION

pint (4, 4, whichmeets comillon I

Boundary r

`Boundarypoint

FIGURE 1-1. Interior and boundary points for a domain D.

FIGURE 1-2.Disconnected regions which fail to satisfy condition 2 for a domain.

A single-valued function in the x, y-plane can be roughly thought of asone or more curves which associate a single value y with each value x. Forexample, the function y= x2/2 shown in figure 1-3 associates the single number4/2 with each point xo. On the other hand, a multivalued function associatesmore than one value of y with each value of x. For example, the functiontan-1 x, which is shown in figure 1-4, associates infinitely many values of ywith each value of x.

In order to avoid ambiguity we shall always suppose that every functionwhich is encountered is single valued unless explicitly stated otherwise.

Consider the equation

F(x, y)=0 (1-1)

3

DIKERENTIAL EQUATIONS

FIGURE 1-3.Singicvalued function y=x7/2.

FIGURE 1-4. Function y----tan-' x.

4

INTRODUCTION

where F has continuous first partial derivatives with respect to its arguments.We say that the function y= f (x) is a solution of this equation if

F[x, fix)] = 0

for all values of x for which f(x) is defined. The solutions y=f(x) of equation!(1-1) arc said to be determined implicitly by this equation and arc calledimplicit functions.

In fact, the implicit function theorem states that for every point (xo, yo)such that

and

F(xo, yo) = 0

aF(xo, yo) 76 0ay (7 -2)

there is/a unique, continuous solution y=f(x) of equation (1- 1) which is de-fined on some neighborhood of the point xo, which sat;3fies the conditionyo=f(xo) and which has a continuous first derivative in this neighborhood.

Now if F(x, y) is a reasonable function, equation (1-1) will be the equationof a curve in the x, y-plane. For example, the implicit relation between x and y

F(x,y) = x y3 =0 (1-3)

is the equation of the cubical parabola shown in figure 1-5. It has an explicitsolution given bye

for all values of x.

The function Aim x in defined by

Y =f(x) m. (MP x)ixi"3 (1-4)

1+1 forsa0tlforx<i)

5


FIGURE 1-5.Cubical parabola x= y3.

Differentiating this equation with respect to x shows that

12=f' (x)=

Hence, condition (1-2) is satisfied at any point A on the curve in figure 1-2which does not coincide with the origin. And, as required by the theorem,equation (1-4) provides a unique solution with a continuous first derivativein some neighborhood of this point. However, condition (1-2) does not holdat the origin. But then the solution (1-4) has an infinite derivative (which iscertainly discontinuous) at this point.

More generally, consider the equation

F(xl,x2, . . ,xn,y)=0

A solution to this equation is a function

Y=fixi,x2, ,xn)

6

(1-5)

with the property that

F[XI Xl . xn,f(xi, x2, . . xn)]*--- 0

INTRODUCTION

for all values of xi, x2, . . xn for which f is defined. In order to obtain ageometric picture of equation (1-5) and its solution, it is convenient to thinkof the variables xi, x2, . xn, y as the coordinates of a point in an (n+dimensional space.

Now suppose that the function F has first partial derivatives with respectto its arguments in some domain D of this space. The implicit function theoremnow shows that for every point (4,4, . . g y°) of D such that

F (x?, . . .4 ) =0

aF(4, . y°) 0

there is a unique continuous solution

Y=fixt, x7, . . xn)

of equation (1-5) which is defined in some neighborhood of the point 4,. . xn, satisfies the condition ya=fig, xi, . . 4,), and has continuous

first partial derivatives in this neighborhood.

1.2 DEFINITION OF DIFFERENTIAL EQUATION

A differential equation is an equation connecting the values of a function,called the dependent variable, the derivatives of this function, and certainknown quantities. If the dependent variable is a function of a single variable(independent) variables, the differential equation is called a partial differentialdifferential equation. If the dependent variable is a function of two or more(independent) variables, the differential equation is called a partial differentialequation.

Thus, an ordinary differential equation is an equation of the form

dy dulF (x,y,17-c, , (1-6)

7


where the positive integer n, the order of the highest derivative which appearsin equation (1-6), is called the order of the equation and F is a function ofthe indicated n +2 quantities.

If the function F in equation (1-6) is a polynomial of degree m in thehighest order derivative dnyldx", we say that it is a differential equation ofdegree m. Thus, the first-order differential equation of degree m has the form 3

F (x, y, y') Mo(x, y) (y')"' Mk(x, Y)(Y' )m-k = 0

The differential equation (1-6) is said to be linear if the dependent vari-able and all its derivatives appear only to the first degree. Thus, the generallinear equation can be written in the form

dny dntv dyao(x)+ a k x ) . .+ao-i(x)±an(x)y+b(x)=0 (1-7)An I dxnI dx

When the function b(x) in equation (1-7) is identically zero, the equation issaid to be homogeneous.

1.3 SOLUTIONS AND INTEGRALS OF DIFFERENTIAL EQUATIONS

A (particular) solution4 of the differential equations (1-6) is any n-times differentiable function f (x) defined on some interval a < x < b (whichmay be infinite) such that equation (1-6) becomes an identity when y and itsderivatives are replaced by f (x) and its derivatives. Hence,

r d.f(x) din 0F (X) (TX den j

for all x in the interval a < x < b. Thus, the function y= sin x is a solutionof the differential equation y" + y=0.

3 We shall frequently write 1' or f' (x) for df (x)Idx, 1" or 1" (x) for d2fldx2, . . ., and po or foo(x) in place

of &fide for n =21,2, . .

'The term "solution" was first used by Lagrange (1774).'There are also solutions known as weak solutions which satisfy the differential eq. (1-6) only in a certain average

sense. We shall not pursue this topic further here. The interested reader is referred to ref. 1 for an elementary treat.ment and to ref. 2 fora more advanced discussion.

8

Now consider the equation

F(X, Y9 Ci1 C29 .9 0

INTRODUCTION

(1-8)

in which x and y are variables, and c1, c2, . ., cn are independents arbitraryconstants.? We shall suppose that equation (1-8) possesses solutions for atleast certain values of the constants c1, c2, , Cn.

Upon differentiating equation (1-8) with respect to x, n times in succes-sion, we obtain

OF OFax ay

a2F 32F a2F ., aFax2 axay (Y1)- +-ay Y" 0

anF aF+. . + y " = 0aY

If the n constants can be eliminated between these n equations and equation(1-8), we can, upon carrying out this elimination, obtain a differential equation 8

G[x, 3" or" .,P)]=0 (1-9)

It is clear, from the manner in which equation (1-9) was obtained, thatevery solution of equation (1-8) satisfies9 equation (1-9). The function Fis therefore sometimes referred to as a primitive of equation (1-9). In manyinstances the solutions of equation (1-8) for y as a function of x can be ex-pressed as formulas of the form

This means that eq. (1-8) is not expressible in terms of fewer than n constants. For example, y2 --x2 + --c2depends effectively only on the single constant c= c2.

7 We assume that all functions are differentiable as many times as is necessary for the argument and that aFlayis not identically zero.

"In practice it usually will not be possible to carry out all the algebraic operations which are necessary to obtainan explicit formula for eq. (1-9).

9 However, as we shall see in the following example, the converse of this statement is not necessarily true.

9


y=f(x, c1, c2, . . cn) (1-10)

which are parameterized by the n constants ci, c2, ., cn. We thereforeanticipate that, for many nth-order differential equations, there exist formulasof the form (1-10) each of which contains n independent constants c1, cnand provides a solution to its corresponding equation for every set of valuesof these n constants for which the formula makes sense.° Such a formula,if it exists, is said to be a general solution of the equation.

This terminology should not be interpreted to mean, however, that everydifferential equation possesses a single general solution from which everysolution to the equation can be obtained by suitably choosing the values ofthe constants. This is only true when certain restrictions are imposed on thedifferential equation (1-6).

For example, consider the equation "

F (x, y, c) c2-n[x_ c(n-1)]2 yn 1=0 for n=1 orn=2 (1-11)

whe-e c is an arbitrary parameter which can take on any real value. By differ-entiating this equation with respect to x we find that

F s+ Fy= 20-1[x 671-11+ = 0 (1-12)

First, consider the case where n= 1. Then upon eliminating c betweenequations (1-11) and (1-12) we obtain the first-order linear equation

2(1 y)o+ x-1

Now, for each value of c, equation (1-11) possesses a single solution

y= 1 c(x 1 )2 for co < x < co

1-13)

(1-14)

This must, therefore. be a general solution to the differential equation (1-13).

'° It is usually necessary to restrict the range of the c's and of x in order to avoid imaginary expressions and otherdegeneracies.

" When n 1, the equation represents a family of parabolas with vertices at the point x= 1, y= 1; and whenn = 2, it represents a family of unit circles with centers on the x-axis.

10 '41

INTRODUCTION

And it happens that every solution to equation (1-13) can be obtained fromequation (1-14) by a suitable choice of the constant c.

Next, consider the case where n = 2. Then upon eliminating c betweenequations (1-11) and (1-12), we obtain the differential equation

y2(y'2+ 1) ( -15)

For each value of c, equation (1-11) now possesses the two solutions

y= V1- (x-c)21(x-c)2 f

for Ix c I 1

12

(1-16)

Hence, both of these formulae are general solutions to equation (1-15). Thus,there is no single formula involving only a single parameter from which everysolution can be obtained." This is not surprising since equation (1-15) canbe written as

4 d(12-(2)2+ y2 1 = 0x(1-17)

and since only y2 appears in this equation, it is clear that if y=f(x) is a solution,so is y=-f(x).

Even though equations (1-16) are the only general solutions to equation(1-15), it is still not possible to obtain every solution to the differential equa-tion from these two formulas. Thus, equation (1-15) also possesses the twosolutions

y=+ 1

1

which not only cannot be obtained from either of equations (1-16) but do noteven satisfy the primitive equation (1-11). They are called singular solutionsof equation (1-15) (see section 1.5) and are tangent to every solution obtainedfrom the general solutions.

12 For any positive real variable x, Vi will always denote the positive square root of x.13 Notice that we can combine the two eqs. (1-16) into a single formula y= (-1)" VI (xc)/ for n=0, 1,

but this solution involves the two parameters n and c.

11

988-942 0 - 73 - 2


We have shown how, an nth-order diffemaial equation can be obtainedby successively differentiating an equation of the type (1-8). In fact if, for agiven differential equation, we can find an implicit relation of the type (1-8)such that every solution of the differential equation is also a solution of thisequation, we can for practical purposes consider the differential -equationsolved. Thus, the process of finding a solution to a differential equation mightbe thought of as reversing the process of obtaining a differential equationfrom its primitive. The first step of this inverse process when applied to thenth-order differential equation (1-6) (if it can be carried out) leads to a differ-ential equation of order n 1 which involves a single arbitrary constant c.And if this equation can be solved for c, it can be written in the form

H (x , y , yf y(" -0) = c (1-18)

More generally, if for any given function H every solution of equation (1-6)satisfies an equation of the form (1-18) for some value of c, this function(and sometimes eq. (1-18) itself) is called a (first) integral of equation 14 (1-6).Geometrically, this means that an integral H is constant along every solutioncurve y=-- f (x) of the differential equation.

1.4 SOLUTIONS TO NORMAL EQUATIONS

If we try to find a solution to equation (1-15), say

y= f (x)

which satisfies the initial condition f(1) = 0, we find from the first equation(1-16) that

y=f

and from the second equation (1-16) that

y=f (x) = V1 x'

"Since integration is the inverse of differentiation, the preceding remarks show why the process of solving adifferential equation is sometimes referred to as integrating the equation.

12

INTRODUCTION

Therefore, equation (1-15) has two solutions which satisfy the conditiony =0 at x= 1. However, we usually expect the solution of any first-orderdifferential equation arising from a physical problem to be uniquely determinedby a single initial condition. It is, therefore, necessary to find what conditionsmust be imposed on a given differential equation if our expectations are to bejustified. To this end suppose that equation (1-6) can be solved for its highestorder derivative. It can then be written as an equation of the form

yoo= G(x, y, y', .00-0) (1-19)

Any ordinary differential equation which has the form of equation (1-19) iscalled a normal differential equation or is said to be in normal form.

We are interested in finding conditions which ensure that equations ofthis type possess unique solutions satisfying the initial conditions

x=x0, y=Yo, Yf =Y1, ,Y("-1)=Y71-1 (1-20)

where Yo, . . . , Y I are constants. The fundamental theorem for nth-orderdifferential equations states that this is always the case, at least locally,provided that certain restrictions are imposed on the function G. In order tostate this theorem in a precise way, let us for the moment treat the variablesx, y, y', . . y(n -') as independent. Then if there exists a positive number 8such that the functions

aG (32G "- iGay, a y2, , ayn_l

are defined and continuous 15 for all values of the variables x, y, y',yo- which lie in the range

Ixxol < 5, IYY01 < 8,13" < a, . .

(1-21)

the fundamental theorem states that equation (1-19) possesses a solutiony=--f(x) which satisfies the initial conditions (1-20) and is defined on some

" Notice that only the partial derivatives of C with respect to the variables y, y', . . , y(") need be continuous,but the partial derivative with respect to x need not even exist.

13


interval containing the point x= xo, say

a < x < b (1-22)

In addition, this solution is unique. This means that if g(x) is any other solutionto equation (1-19) which satisfies the initial conditions (1-20), then

f(x) = g(x)

for all x in the interval's (1-22).Notice that we have only asserted that the interval (1-22) exists; we have

not given any idea of its size.'7 That is, there is no relation given between a, b,and S. The interval might be exceedingly small. However, we shall give ameasure of the size of this interval for an important special case. Thus, if thepartial derivatives (1-21) are defined and continuous for all values of y, y',. . . , y(n-1) and for all values of x in some interval which contains the pointx=x0, say

a < x < p

and if there exists a single constant K such that

Vii)K for i = 0,1,2, . . . , n1

(1-23)

for all y, . . . , yo- and all x in the interval (1-23), the solution y=f(x)which satisfies the initial conditions (1-20) exists and satisfies the differentialequation for all x on the interval (1-23).

For example, it is easy to verify that the equation

y' = G (y) = 1 + y2

has the solution y= tan (x+c ). Since tan x becomes infinite at x= ±ir /2, it

16 Actually, the same conclusions can be reached even when somewhat weaker conditions are imposed on the func-tion Gin eq. (1-19). Proofs of the fundamental theorem can be found in refs. 3 to 5.

'71n fact, a measure of the size of this interval can be given in terms of certain bounds on the function G. See, forexample, ref. 4, theorem 8, p. 118.

14

INTRODUCTION

is clear that the solutions to this differential equation are defined only on inter-vals whose lengths are at most 71", even though G and 0Glay are defined andcontinuous everywhere. This occurs because there is no constant K which islarger than aGlay= 2y for all values of y.

Although we normally expect the solution of an nth-order differentialequation to be uniquely determined by specifying n initial conditions (or theequivalent), we have seen by example that specifying these conditions is notnecessarily sufficient to uniquely determine the solution to every nth-orderequation. The preceding discussion indicates that the extraneous solutionsmust arise either because the differential equation (1-6) is not in normalform (i.e., solved in a unique way for the highest order derivative) or because,even if it is in the normal form (1-19), the function G is not sufficiently smoothat some points.

A general solution to equation (1-19), if it exists, contains n "arbitrary"constants. And since, in principle, it is usually possible to solve it equationsin n unknowns, this is consistent with the fundamental theorem which statesthat n initial conditions determine the solution to equation (1-19).

For example, integrating the differential equation

yfi=x2 (1-24)

twice yields the general solution

X4y12

=-+ CiX±C2 (1-25)

containing two arbitrary constants c1 and c2. The constants are uniquelydetermined by the initial conditions

since the two equations

x= xo,y =Yo,Y' =Y1

PX

12,

C1X0+ C2

x?

3

, ,

-t- ci=

15


can be solved for c1 and c2 to obtain

C2 = Yo XoY =4

More generally, suppose that by some formal process of integration wehave found m general solutions

Y=./i(x, ci,c2, . . .,cn) i=1,2, . . .,m (1-26)

of the differential equation (1-19) and that every solution of this differentialequation can be obtained from these formulas by properly choosing the con-stants. Then for every set of values of the n constants xo, Yo, Y1, 7 Ynifor which G satisfies the conditions imposed in the fundamental theorem,there must be one and only one value of i for which it is possible to solve then equations

Yo=fi(xo, ci,c2, . . cn)

dfiY=. c 1, c2, 7 Cn)

dx

dn''ffYn 1 dXn-1

(X0, Cl, C2, , Cn)

for the n constants ci, c2, . . , Cr:-

For example, the function

y=f(x, c) V1+ces

satisfies the differential equation

16

(1-27)

1 1\y'=F(y).7=

INTRODUCTION

(1-28)

for all values of c for which the radical exists. And for this equation the partialderivatives (1-21) are always continuous except when y= 0. Hence, the funda-mental theorem shows that there must be a solution which satisfies the initialcondition

x=0

However, there is no possible choice of c in equation (1-27) for which f(0,c) = 1, which shows that it is not possible to obtain every solution to equa-tion (1-28) from the formula (1-27). However, the function

N/1 cex (1-29)

also satisfies equation (1-28) for all values of c for which the radical exists.Equations (1-27) and (1-29) taken together will now provide a unique solutionfor each initial condition which does not involve y= 0. But these formulas willprovide two solutions satisfying each initial condition involving y= 0, wherethe partial derivatives (1-21) are not continuous. However, the fundamentaltheorem provides no information about solutions which pass through pointswhere the partil derivatives (1-21) are not continuous.

Specifying initial conditions is only one of many ways of determining thevalues of the arbitrary constants which appear in the general solutions of adifferential equation. The most common alternative is to require that thesolution and its derivatives satisfy certain conditions at both ends of someinterval, say xi = x x2, within which this solution is being sought. Theseconditions are called boundary conditions.18

For example, we might require that the solution to equation (1-24) satisfythe boundary conditions

18 The term "initial conditions" arose in conjunction with problems in mechanics which have time as the independ-ent variable and have conditions imposed at some initial time. The term "boundary conditions" arose in conjunctionwith problems involving physical distance as the independent variable with conditions imposed at the boundaries of aphysical region.

17


y=yi atx =x1

y=y2 atx=x2

Then substituting equation (1-25) into these conditions shows that

4Xical + C2=--t-

12-

4X2y2=12 + cix2+ C2

And since these equations have the unique solution

ci Y2 4 4xi x2+ 12(x1 -x2)

c2 =x2yi-xiy2 , xix2(xg--xl)12(x2-xi)

we conclude that the two boundary conditions uniquely determine the solutionof e.quation (1-24). There are a number of theorems which give conditionswhich; 'if ::satisfied; will ensure that the solution of an nth-order equation willbe uniquely determined by n boundary conditions. However, since these con-ditions are fairly complicated, we will not state any of these theorems herein.

1.5 SOLUTIONS TO EQUATIONS NOT IN NORMAL FORM-SPNGULAR SOLUTIONS

We have shown that the nth-order normal differential equation possessesa solution which is uniquely determined by n initial conditions, provided that theconditions imposed in the fundamental theorem are satisfied. In order to usethis result to obtain information about the solutions of a differential equationwhich is not in normal form, it is necessary to solve this equation, at least inprinciple, for its highest order derivative.

First, consider the general nth-order equation of the first degree

U(x,y, . . .,y01-19y(n)+V(x,y, . . .,y0i-1))=0 (1-30)

This equation can be written in the normal form

18

G(x, y, , y(n-i)) = V (x, y, . y(n -1))

U(x, y, y(n -1))

INTRODUCTION

(1-31)

for all values of x, y, . . y(n-1) for which U is not equal to zero. We shallsuppose that U and V are defined and possess continuous first partial deriva-tives for all values of x, y, . . y(n-1). Then the function G will possess con-tinuous first partial derivatives for all values of x, y, . . y(n-1) at whichU 0 0. Hence, the fundamental theorem shows that equation (1-30) will possessa unique solution satisfying any set of n initial conditions x = xo, y= yo, .

y(n-1)=-_- Yn-i for which U(xo, Yo, Yni) 0 0. Each solution of equation(1-31) will satisfy equation (1-30); and if U is never equal to zero, every solu-tion of equation (1-30) will satisfy equation (1-31). Hence, in this case, equa-tions (1-30) and (1-31) are equivalent. However, suppose that the function 11vanishes for certain values of its arguments. If equation (1-30) possesses asolution y=f(x) such that for certain values19 of x

U( , f (x) , . , Pn-1)(x))= 0 (1-32)

then y=f(x) will not necessarily satisfy equation (1-31) for these values"of x. It is called a singular solution of equation (1-30).

For example, the first-order equation

37' (Y2Y) = 0

can be written in the normal form

y2y - = y- 1

The normal equation (1-34) has the general solution

y= 1+ cex

(1-33)

(1-34)

"Thus y= f(x) satisfies both eq. (1-30) and the (n 1)st-order differential equation U(x, y, . . yo'-'))=0.20 Even if the numerator of G vanishes in such a way that it is possible to define G(x ,f (x), . . fth-1) (x)) at these

values of x.

19


and this is also a general solution of equation (1-33). However, equation(1-33) also has the solution

y=0 (1-35)

but equation (1-34) does not. Since the coefficient of y' vanishes at y =0,equation (1-35) is a singular solution of equation (1-33).

Now consider the general equation (1-6) and let us temporarily treatthe quantities x, y, . . y(n) as independent variables. Suppose that F hascontinuous first partial derivatives with respect to its arguments. Then theimplicit function theorem (given in section 1.1) shows that equation (1-6)has several (possibly infinitely many) solutions of the form

1)=G(x, y, .00-0) (1-36)

where the function G is continuous and has continuous first partial derivativesfor all values of its arguments for which it is defined. One of these solutionswill satisfy equation (1-6) for each set of values of x, y, . . y(71) for whichequation (1-6) holds and

aFay(n)

(x'y, . . P)) 0 0 (1 -37)

However, equation (1-36) is a normal differential equation; and therefore thefundamental theorem shows that it possesses a unique solution satisfyingany set of initial conditions x = xo, y = Yo, . . y(n -') = Yn-i for which G(xo,Yo, , Yo--1) is defined. All the solutions of the normal equations of the form(1-36) obtained from equation (1-6) will satisfy equation (1-6). If aF/ay(n)is never equal to zero for any values of x, y, . . y(n) for which equation(1-6) holds, every solution of equation (1-6) will satisfy one of the equations(1-36). Hence, in this case, equation (1-6) is equivalent to the set of normalequations of the form (1-36). However, suppose that

aFap)(x,y,y', y(n))=0 (1-38)

for certain values of x, y, . . yo). If equation (1-6) possesses a solution

20

y= f (x) such that for certain values 21 of x

aFapo (x,f(x), . ,f(n)(4)=0

INTRODUCTION

(1-39)

then for these values ex the implicit function theorem no longer guaranteesthat there will be normal equations (satisfying the conditions imposed in thefundam, al theorem) which are satisfied by the solution y= f (x) of equation(1-6). This solution is, therefore, called a singular solution of equation (1-6).

For example, the equation

has the two solutions

F (y, y' ) y'2 +1 -y=0 (1-40)

Y = (Y) = VT (1-41)

= G2 (y) -=- -VT-TT (1-42)

where G1 and G2 have continuous derivatives for all values y, y' for which equa-tion (1-40) holds and for whirl'

a=2y, #0

Now equation (1-41) has the solution

(1-43)

y=l+ 1T(x -c)2 forx c (1-44)

and equation (1-42) has the solution

1y= 1+4

(x- c)2 forx c (1-45)

21 Thus yr-- f (x) satisfies both the differential eq. (1-6) and the differential eq. (1-38).

21


y

7

I

x > c

-3 -2 -1 0 1 3

FIGURE 1-6.Solutions of equations (1-41) and (1-42).

For each value of c, these two solutions are the two branches of the same parab-ola. They are shown in figure 1-6. The solutions of equation (1-41) are shownas solid curves. The solutions of equation (1-42) are shown dashed. Noticethat one solution to equation (1-41) and one solution to equation (1-42)pass through each point (such as point A) lying above the line y=1, which isconsistent with the fundamental theorem.

Equation (1-43) shows that the points where y' = 0 are exceptional.But, for those values of y and y' which satisfy equation (1-40) , this can occuronly when y= 1. However, when y= 1, dGil dy and dG2 /dy become infinite;and therefore the fundamental theorem does not apply along this line. Equa-tions (1-41) and (1-42) (and, therefore, also eq. (1-40)) possess the softition

y =1

which lies along this line and which cannot be obtained for any choice of cfrom either equation (1-44) or (1-45). Thus, y =1 is a singular solution.Notice that there are three solutions of equation (1-40) passing through eachpoint on the line y= 1.

Now consider the first-order differential equation-

F (x, y, y') = 0 (1-46)

The singular solutions of this equation, if they exist, must also satisfy theequation

22

aFay '(x, r,y')=0

INTRODUCTION

(1-47)

When y' can be eliminated between equations (1-40) and (1-41), we obtainthe equation

g(x, y) = 0 (1-48)

Since the singular solutions simultaneously satisfy equations (1-46) and(1-47), they must also satisfy equation (1-48). Thus, every singular solution ofequation (1-46) can be found by solving equation (1-48). However, the con-verse is by no means always true. Since the symbol p is often used to denotey', equation (1-48) is sometimes called the p-discriminant equation and thecurve described by this equation is sometimes called the singular locus.When attempting to find all solutions of a first-order equation, the solutions tothe p-discriminant equation should be checked to see if they also satisfy thedifferential equation.

For example, consider the first-order quadratic equation

F(x,y,y1)=Ay12+By' +C=0

where A, B, and C are functions of x and y. Upon eliminating y' between thisequation and the equation

aF=---- 2Ay'+B= 0a'

we find that the p-discriminant equation is

B2 4AC =0

Thus, for the first-order equation

F (x, y, y1) = xy' 2 3yy' + 9x2= 0

the p-discriminant equation is

y2= 4x3

(1-49)

23


which has the two solutions y=±_.2x312. Direct substitution shows that boththese solutions satisfy equation (1-49); and, therefore, they are both singularsolutions.

1.6 LINEAR EQUATIONS

In this section we present a number of important special properties whichare possessed by the solutions of linear equations. The general form of thenth-order linear equation is given in equation (1-7). If f i(x) and f2(x) areany two solutions to a linear homogeneous differential equation and c1 andc2 are any constants, the function

ctfi (x) + c2f2(x) (1-50)

is also a solution. This linear, superposition principle is extremely importantin analysis. It can easily be extended to include linear combinations of anynumber of solutions. Notice that the homogeneous equation always possessesthe trivial solution y= 0.

The equation obtained from a particular linear equation of the form (1-7)by setting b(x) equal to zero is called the associated homogeneous equation

of this equation. Any solution to the associated homogeneous equation iscalled a homogeneous solution of the equation. If f(x) is any solution of equation(1-7) and if fli(x) is any homogeneous solution of this equation, the function

f(x) + cfH(x)

is also a solution. Of course, the superposition principle holds only for thehomogeneous solutions of a linear equation.

Notice that the linear equation (1-7) can always be written in the normalform (1-19) with

al an-i , anG ,(n -1) _ . . y --y--- bao ao ao

(1-51)

for all values of x where the coefficient ao(x) is not equal to zero. It followsthat for these values of x

aG an_i(x)y(i) ao(x) for j= 0,1,2, . . n-1

24

(1-52)

INTRODUCTION

Now suppose that the coefficients ao, al, . , an, b of equation (1-7) areall continuous in some finite 22 interval

a /3 (1-53)

Then if ao(x) does not vanish at any point of this interval, it will always bepossible to find a constant K such that

an -' (x) for j= 1, 2, 3, . . n 1ao(x)

and for all x in the interval (1-53). It can also be seen from equations (1-51)and (1-52) that the partial derivatives (1-21) are continuous for all values ofy, y', . . yo-1) and all values of x in this interval. Hence, in this case, Gnot only satisfies the conditions imposed in the fundamental theorem, but itsatisfies the more restrictive conditions given immediately following it.

We therefore conclude that for linear equations the fundamental theoremcan be stated in the following way:

Let the coefficients ao(x), ai(x), . . an(x), b(x) of equation (1-7) be con-tinuous on some interval a < x < p which contains the point xo, and supposethat the function ao(x) does not vanish in this interval. Then for any real numbersYo, . Yn there is a unique solution y = f(x) of equation (1-7) satisfying theinitial conditions

y(xo)= Yo,y'(xo) =Y1, . .,Y(ni)(x0)=Yni

And this solution satisfies the differential equation on the entire intervalcx < x < g.

If the linear equation (1-7) has coefficients which are continuous in someinterval, the points in that interval where the coefficient ao(x) vanishes arecalled singular points of the equation. We shall have more to say about suchpoints subsequently.

A set of functions g1 (x), g2 (x), . . . , gfl(x) is said to be linearly inde-pendent in an interval if it is impossible to find constants ci, cn whichare not all zeros such that the expression

22 This means that neither a nor is equal to infinity.

25


(x) c2g2(x) + . . . cng, (x)

is equal to zero at all points (i.e., identically) of the interval. A set of functionswhich is not linearly independent is said to be linearly dependent.

If the set of functions gi (x), . . gn(x) is linearly independent, no oneof these functions can be expressed as a linear combination of the others.Hence, if ai 0 0, it will not be possible to express the function

f (x) = (x) a2g2(x) . . . + ang.(x) (1-54)

in the form

f (x) =-- eig i (x) + e2g2(x) + . . . + (x) + (x) + . . . + engn(x)

in which the function gi(x) no longer appears. Thus the constants appearingin equation (1-54) are independent."

For example, the functions g1(x) = 2x 5 and g2 (x) = 6x 15 are notlinearly independent since

(x) + c2g2(x) = 0

when c1= 3 and c2 = 1. However, the functions

g1(x) = 2x 5 and g2(x) = 2x + 5

are linearly independent.Let 8,1 , g2, . . gn be a set of (it 1)-times continuously differentiable

functions. The determinant

26

PV(gi,g2,

g1 g2 gn

in-1) gtn-1) gin-1). .

23 This means that eq. (1-54) is not expressible in terms of fewer than n constants.

INTRODUCTION

is called the Wronskian of these functions. Now suppose that the functionsg2, . gn are linearly dependent on some interval. Then there exist

constants cl, c2, . . c, not all zero such that

cigi(x) c2g2(x) + . . . + cng,,(x)= 0

at every point of the interval. Hence, we can differentiate this expressionn-1 times to obtain

cig;(x) + c2g2 (x) +. . .+ cng.' (x) =0

cig1n-1)(x) + c2g2-1)(x) + . . . + cngnn-1)(x) = 0

This giVes us a set of n linear equations for the n nonzero constantsc2, . . cn. If these equations are to be solvable for the constants at all

points of the interval, the Wronskian must vanish at all points of this interval.Hence, we can conclude that if the set ga, g2, . ga of (n 1) -times continu-ously differentiable functions is linearly dependent on the interval a x #,

. the Wronskian of these functions must vanish at all points of this interval.Another way of saying this is that if the Wronskian of a set g1, g2, . gt,

of (n 1)-times continuously differentiable functions does not vanish at everypoint of the interval a x p, these functions are linearly independent on thisinterval.

Thus, the Wronskian of the functions (discussed in the previous example)g1(x) = 2x 5 and g2(x) = 6x 15 is

2x 5 6x -15

2 6

= (2x-5)6-2(6x-15) =0

W (gi g2 ) =

And this shows that these functions are linearly dependent. However, theWronskian of the linearly independent functions gi (x) = 2x 5 and g2 (x) =2x +5 is

27

488-942 0 - 73 - 3


W(gi , g2) =2x-5 2x +5

2 2= 20

which is certainly not equal to zero.Now consider the associated homogeneous equation of equation (1-7)

aoyoo+ + . . . + + any= 0 (1-55)

and suppose that the coefficients ao, a1, . . an are continuous in the interval

a x (1-56)

and that ao(x) does not vanish on this interval. We shall now show that any nsolutions fi(x), . . fn(x) of equation (1-55) are linearly dependent on theinterval (1-56) if, and only if, their Wronskian W(f1, f2, . . fn) vanishes atevery point of this interval.

Another way of saying this is that the n solutions fi(x), . . fn(x) ofequation (1-55) are linearly independent on the interval (1 -56) if, and onlyif the Wronskian . . fn) is not equal to zero at some point of thisinterval.

We have already shown that any set of (n 1)-times continuously differ-entiable functions is linearly dependent on an interval only if their Wronskianvanishes at every point of this interval. Hence, it is certainly true for a set ofn solutions.24 In order to show that, conversely, the vanishing of the Wronskianimplies that the solutions are linearly dependent, suppose that W(A,f2,fn) vanishes at every point of the interval (1-56) and let xo be any point of thisinterval. Then the Wronskian certainly vanishes at this point, and thereforethe system of n equations

24 Recall that the definition of a solution requires that it be n 1 times continuously differentiable.

28

curl (x0) + c..11:2 (x0) + . . + c,f(xo) --= 0

cd:(x0) + cd;(x0) + . . . + c,4 (x0) ---- 0

cif (x0) + c2f (xo) + + c nf ') (xo) =

INTRODUCTION

(1-57)

has a solution El= cl, E2=c2, . . En.= cn such that the constants '61, -625-6 are not all zeros. Hence, we can define a function f(x) by

f(x) etfi(x) + ettl2(x) + . . . + enfn(x) (1-58)

Then it follows_ from the linear superposition principle that f(x) is a solution toequation (1-55) in the interval (1-56). And differentiating equation (1-58)n 1 times shows that

f + -e2f2 + . . . + Eqfn

f' ei/7 + e2fl

f(n-1),- f(in -1) ___ '62f -1) + enpirii -1)

Hence, upon setting x = xo in this system and using equation (1-57), we findthat the solution f satisfies the n initial conditions

f(x0)=fi(x0)= . . . =-_f(n--1)(x0)=0

Since the trivial solution of the homogeneous equation (1-55) also satisfiesthese conditions and the fundamental theorem shows that there is only onesuch solution, we therefore conclude that f(x) is equal to zero at every pointof the interval (1-56). Therefore,

Etfi(x) +e2f2(x)+ . . . +671fn(x) =0

29


at every point of the interval (1-56). But since the constants el, -e2, .,en are not all zero, this implies that the set fi, f2, . . .,fn is linearly dependentand therefore proves the assertion.

We shall now show that the linear homogeneous equation (1-55) alwayshas n linearly independent solutions on the interval (1-56). To this end noticethat for any point xo of this interval the fundamental theorem shows that equa-tion (1-55) has n solutions fl(x), f2(x) , . . . , fn(x) which satisfy the initialconditions

(x0) =1

A' (x0) =./1" (x0) = =fin-i) (x0) =0

and for r= 2,3, . . .,n

fr(x0) = f ; (x0) = . . . =f r-2) (x0) = 0

firr -0 (x0) 1

t.r)(xo) = firr+')(xo)= . . .=Acn -1) (xo)=0

It is easy to see from these conditions that the Wronskian of these n solutionshas the value unity at the point xo. Thus, it is not zero at every point of theinterval a-- 5.x--5. fi, and we can therefore conclude that these n solutions arelinearly independent.

For example, the second-order differential equation

dx2 Ir-=`'

has the two solutions y= sin x and y= cos x.The Wronskian of these solutions is

sin x cos x

cos x sin x

=sin2xcos2 x=-1 0 0

TV(sin x, cos x)

30

INTRODUCTION

Hence, the solutions are linearly independent.If fi, f2, . . fn are any n linearly independent solutions of equation

(1-55) on the interval (1-56), and if h(x) is any other solution of equation(1-55) on this interval, there exist constants Ci, c2, . . .,cn such that

h(x)="elfi(x)± E2f2(x)+ . . .+E0f0(x) (1-59)

on the entire interval.This means that every solution to the equation on this interval can be

obtained from the general solution

f(x, c 1, c2, . . , cn) =cifi(x) + c2f2(x) + . . + cnfn(x)

In order to prove this assertion, notice that since fi, . . fn are linearlyindependent, W(fi, f2, fn) cannot be zero at every point of the interval(1-56). Hence, there exists a point, say xo, for which the Wronskian is not zero.The n equations

h(xo) =eifi(xo) + 62.1.2(xo) + . +6o.fo(xo)

h' (x0) = Elfi(x0) + E2f2(xo) + . . + enn(x0)

(1-60)

h(o-i)(xo) din- 0(x°) 4. -62R-1)(x0) + up-1) (x0)

can therefore be solved for the n constants 7---1, -;" 29 1 Fan. Hence, the linearsuperposition principle shows that the function g(x) defined by

g(x) = di(x)+6zf2(x) + . . . + Enfn(x) (1-61)

is a solution to equation (1-55) on the interval (1-56). And it follows fromequation (1-60) that

h (x0) = g(xo) (xo) = g' (x0), . . , h(n-1)(x0) = en- - 0 (x0)

But since the fundamental theorem shows that there is only one solution

31


satisfying a set of n initial conditions, we can conclude that g(x) = h(x) forall x on the interval (1-56). Equation (1-61) now shows that equation (1-59)holds, and this proves the assertion.

It is easy to see from these results that equation (1-56) cannot possessn + 1 linearly independent solutions on the interval (1-56).

Now consider the general linear equation (1-7) and suppose that thecoefficients ao(x), al(x), . . a(x), b(x) are continuous on the interval

a x p (1-62)

and that ao(x) does not vanish at any point of this interval. Let fo(x) be anyparticular solution of equation (1-7); and let f,, f2, . . . , fo be any n linearlyindependent solutions of the associated homogeneous equation. Then if h(x)satisfies equation (1-7) on this interval we can find n constants c,, . .,such that

h(x) = Elf; (x) + . . . + jitf(x) + fo(x) (1-63)

on the entire interval.This means that every solution on the interval can be obtained from the

general solution

f(x, c1, c2, . . . , cn) = ctfi (x) + czf2(x) + . . + cnfn(x) + fo(x) (1-64)

In order to prove this assertion, notice that h(x) fo(x) is a solution to the as-sociated homogeneous equation of equation (1-7). Hence, we can find constants

1 1 -629 ' 9 &n such that

h (x) fo(x) e (x) + "e2f2(x) + . . . + (x)

This shows that equation (1-64) holds and proves the assertion.The general solution of the associated homogeneous equation

c tfi (x) + czf2(x) + . . . + c7f(x)

which appears in the general solution (1-64) of the nonhomogeneous equation(1-7) is called the complementary function. And any set of n linearly inde-

32

INTRODUCTION

pendent homogeneous solutions is known as a set of fundamental solutionsto the homogeneous equation.

For example, the complementary function of the second-order equation

(1-65)

is ci cos x+c2 sin x where cl and c2 are arbitrary constants. A particularsolution to this equation is y=x. Then

y= c, cos x+c2sinx+x

is a general solution of equation (1-65), and any other particular solution ofthis equation can be obtained by assigning particular numerical values to theconstants.c., and c2.

33/1V

CHAPTER 2

First-Order Equations

In this chapter a number of elementary techniques for obtaining explicitsolutions for certain types of first-order equations are developed. Theseequations are chosen for consideration because they are simple enough tobe solved explicitly and, at the same time, occur frequently enough in practiceso that they are worth solving. As it happens, these equations are all of thefirst degree.

2.1 SYMMETRIC NOTATION FOR EQUATIONS OF FIRST DEGREE

The first-order differential equation of the first degree is an equation of theform

y'N(x, y) + M (x, y) =0 (2-1)

We shall suppose that M and N possess continuous first partial derivatives insome domain D of the x, y plane. If N(x, y) were nonzero at every point ofD, we could write equation (2-1) in the normal form

M(x, Y)yi =fix, Y) N (x, y)(2-2)

and the fundamental theorem would guarantee that this equation have aunique solution passing through each point of D. Points of D where N(x, y) =--are called singular points of the equation.

In equation (2-1), y is considered as the dependent variable. The solutionsof this equation are functions of the form

35


r=fix)

On the other hand, the equation

dx M(x, y) + N (x, y) =0dy

(2-3)

(2-4)

which is closely related to equation (2-1) possesses solutions which are func-tions of the form

x= g(y) (2-5)

The singular points of this equation occur when M (x, y) = 0.However, in any domain in which neither equation (2-1) nor (2-4) has a

singular point, any solution (2-3) of equation (2-1) can be solved for x to obtaina solution of the form (2-5) of equation (2-4), and conversely. Thus, for manypurposes, it is not necessary to distinguish between equations (2-1) and (2-2).When this is the case, we can intr.. Juce the symmetrical notation

M(x, y)dx+ N(x, y)dy=-- 0 (2-6)

If M(x, y) and N(x, y) are both not zero, we understand this notation todenote either equation (2-1) or equation (2-4). When M(x, y) =0 but N(x, y)0 0, the existence of the solution (2-3) of equation (2-1) is guaranteed by thefundamental theorem, but that of equation (2-4) is not.25 Hence, in the neigh-borhood of a point where M(x, y) = 0 we understand the notation (2-6) todenote equation (2-1). Similarly, when N(x, y) =0 but M(x, y) 0, we under-stand the notation (2-6) to denote equation (2-4). The fundamental theoremdoes not guarantee that either equation will possess solutions at pointswhere M(x, y) and N(x, y) are both zero, and we therefore say that suchpoints are singular points of equation (2-6).

25At such points, dxIdy= co and dyldx= 0; hence, dx /dy is not continuous.

36

FIRST-ORDER EQUATIONS

2.2 EXACT EQUATIONS

The first-order differential equation of the first degree

M(x, y)dx+ N(x, y)dy= 0 (2-7)

with M and N denned and continuous in some domain D, is said to be exactin D if the line integral

[M(x, y)dx + N(x, y)dy]

has the same value for all paths of integration which lie in D and which have thesame end points. This is equivalent to requiring that the integral vanish.aroundevery closed path within D.

It is shown in books on calculus that the differential equation (2-7) isexact if, and only if, there exists a continuously differentiable function 4)(x, y)such that26

M=-84)a cto

and N ay(2-8)

Then the function ct), which is defined only to within an additive constant,is given by

(x,u)0(x, y) = (11fi :x+ Nay) + constant

f(xoo0),

(2-9)

where (xo, yo) is any fixed point and the integral is carried out over any curveF within D which joins the fixed point (xo, Yo) with the variable point (x, y).

Suppose that equation (2-7) is exact. Then it follows from equation (2-8)that the total derivative of (/) with respect to x along any direction is

cid) ad) +L =M+ Ndx ax ay

Similarly, the total derivative with respect toy is

(2-10)

26Alternatiyely, we can say that M dx+ N dy is an exact differential of the function cf, or that dch = M dx+ N dy.

37


_ao dx dxdy ay ax dy dy

Since the differential equation (2-7) denotes either the equation

or the equation

M+y'N=0

dxN+dy M=0

(2-11)

equations (2-10) and (2-11) show that is constant along every solutioncurve of equation (2-7). Hence, 4) is an integral of equation (2-7). However,for a first-order equation, finding its integral is equivalent to finding its solu-tion in implicit form. Thus, an exact equation can effectively be solved simplyby carrying out the integral (2-9) along any convenient path.

It is easy to see by differentiating equations (2-8) and interchanging theorder of differentiation that, if M and N are continuously differentiable, anecessary condition for equation (2-7) to be exact is that

aM,_aNay ax

(2-12).

Now suppose that the domain D has no holes. That is, it is similar to thedomain shown in figure 2-1(a) but not the one shown in figure 2-1(b). Such adomain is said to be simply connected.

Let I' be any closed curve within D and let R be the region enclosed byI', as shown schematically in figure 2-2. Then Green's theorem

[M dx+ N dy] = f (11-V dx dyax 1 y

shows that condition (2-12) is also a sufficient condition for exactness. If Dwere not simply connected, condition (2-12) would not be sufficient to ensurethat the line integral will always vanish along a closed path such a'4 r in figure2-2.

38

(a)


(b)

(a) Domain has no holes. (b) Domain has a single hole.

FIGURE 2 -1. Singly and multiply connected domains.

FIGURE 2-2.Path of integration within D.

We shall always suppose that the conditions of differentiability and con-nectedness given in the preceding paragraphs are satisfied and therefore thatequation (2-12) is the necessary and sufficient condition for equation (2-7)to be exact.

For example, consider the equation

x22

xyy2

dxx22

y+ y

x2dy= 0

39


Then

and

y

(x0, y)

x

FIGURE 2-3.Path of integration for finding integral.

am a ( 2x- y2 -x2 4xyay ay x2 + y2 (x2 + y2 ) 2

aN a (2y+ x y2 x2 -4xyax axx2+ y2) (x2 + y2) 2

Hence, equation (2-12) holds and the differential equation is exact. Therefore,carrying out the integral (2-9) along the path depicted in figure 2-3 shows that

Y 2y+ xo fx2 + y2

x 2x -v(13(x, y) =1

+ y2dy+ dx

=1n x + y2) tan-1-x+ constant

is an integral of the equation.Unfortunately, the majority of first-degree equations encountered in

practice are not exact. It is therefore natural to ask whether there exists afunction X (X, y) such that the equation

X(M dx+ N dy) = 0 (2-13)

40


(obtained by multiplying both sides of eq. (2-7) by A.) is exact. It can be shown(ref. 3, p. 27) that such a function always exists, provided only that equation(2-7) possesses exacdy one general solution. It is called an integrating factor.Integrating factors are certainly not unique for, if X is an integrating factor ofa given first-order equation, so is cX for any constant c.

Now if 'X is an integrating factor for equation (2-7), the condition (2-12)for exactness must hold for equation (2-13). Hence,

or

a (xm) _a (xN)ay ax

ax ax (8M aNM --N + X ay ax) 0ay ax (2-14)

Thus, X. is the solution of a linear partial differential equation (see section 3.3)which is usually more difficult to solve than the original ordinary differentialequation. However, since it can be shown that every solution of equation(2-14) is an integrating factor of equation (2-7), it is only necessary to find asingle particular solution to equation (2-14).

Although, in general, finding an integrating factor is quite difficult, it issometimes possible to accomplish this simply by inspection. For example,upon multiplying the equation

[f(x) + y]dx x dy= 0

by 1/x2 we obtain the equation

1 1[f(x)+ Ada, --xdy= 0

And since a lay[ (f+ y)/ x2] =1 /x2 and a/ax(-1/x) = 1/x2, this equation isexact. Then carrying out the line integral (2-9) shows that the differentialequation possesses the integral

f121 dxx2

41


2.3 EQUATIONS OF THE TYPE g(y)y' =f(x)-Fh(x)G ff(x)dx f g(y)dy)

Many of the differential equations which can be solved by classicalmethods are special cases of the differential equation

g(y)dy+[f(x)+h(x)G f(x)dxf g(y)dy)] dx= 0 (2-15)

where g(y), f(x),h(x), and G can be any functions of their arguments. In orderto obtain a solution to this equation put

U= f(x)dx g(y)dy

Then dU= f(x)dxg(y)dy and equation (2-15) can be written as

dU+h(x)G(U)dx=0

It is easy to see that 1 /G(U) is an integrating factor for this differential equationand therefore that the equation

dUG(U)+h(x)dx=0

is exact. Hence, it follows from equation (2-9) that

I fix)dx I g( Y)dY 1dU+f h(x)dx= constantG(U) (2-16)

is an integral of equation (2-15).

We now show that many of the first-order normal equations, which aresolvable by the classical methods, can be obtained by specializing the functionsf, g, h, and G in equation (2-15) and, therefore, that the solutions of theseequations (actually the integrals) are all given by equation (2-16).

42


First put G(U)= 1, and h(x) =0. Then equation (2-15) becomes thegeneral separable equation

g(y)dy+ f (x)dx= 0

and equation (2-16) shows that its integral is simply

f(x)dx. f g(y)dy

Now put G(U) = (al (3)eulaH(e-u) 1, g(y) y, f(x)=h(x)Then equation (2-15) becomes

dy= x(010-1Hx0

dx (2-17)

which is the isobaric (or one-dimensional) equation. In this case, the integral(2-16) becomes

or

In (a43/ya) dU In xo= constanta eu/aH(e-u) 1

dV In xo= constantVI-010H (V) -V

If we put a= 1 in equation (2-17), we obtain the homogeneous equation

dy= H (-)x) dx (2-18)

And equation (2-16) shows that its solution is

fulx dVH (V) V ln x= constant

488-942 0 - 73 - 4

43


In order to recognize whether a given differential equation has the. form(2-17), it is necessary to be able to determine whether any given function71(x, y) can be expressed in the form

x(01.2)--'HCxsl

(2-19)

A necessary and sufficient condition that 71(x, y) can be expressed in this formis that there exist a number p such that

71(tx, tPy) = tP-'n (x, y) (2 -20)

for all values of t.In order to show that a function which satisfies the condition (2-20) can

always be expressed in the form (2-19), put t= 1/x in equation (2-20) to get

7 (1, x--3i) =x'-Pn(x,

Now choose a number a and define the number f3 and the function H(U) byp = pa and H(U) = 71(1, WI"), respectively. Then

H (5) 71 11, ; y 7 11, ( 1(igla)xt3la = 71 1 9 xi X 7)(x, y)

which shows that n(x, y) can be expressed in the form (2-19).For example, replace y by ytva and x by xt in the function

to show that

71(x, y). x)x2 (a In y In x)

71(xt, = to-oian (x, y)

Hence, this function satisfies condition (2-20), and it can be put in the form(2-19) by introducing the variable fi'x to obtain

44


(1]X X71(x, y) = x( 1/a)-1rra [1

ln Yax

In the important special case where a= /3 (corresponding to the homogeneouseq. (2-18)), condition (2-20) reduces to

71(tx, ty) = 77 (xor) (2-21)

A function of two variables which satisfies condition (2-21) is said to be homo-geneous of degree zero, which, of course, accounts for the name given toequation (2-18). More generally, a function of the n variables xl, x2, . . xn,say n(x1, x2, . . . , xn) , is said to be homogeneous of degree k in the variablesxl, X2, . Xn if it satisfies the condition

tx2, . txn) = t'71 (xi, x2, ., xn) (2-22)

for all values of t.Another well-known special case of (2-15) is obtained by putting

G(U) = e-oc-1)U, h(x) = (x) exp [(k 1) f(x)dx], and g(y) = 1/y

to obtain Bernoulli's equation

dy = [f (x)y + (x) yk]dx

It follows from equation (2-16) that its solution is

(2-23)

exp [(k 1) f(x)dx]

(k 1)y*--1+ f exp [(k 1) J f(x)dx] = constant

(2 -24)

When k = 0, equation (2-23) reduces to the first-order linear equation

dy = [f (x)y + tli(x)]dx (2-25)

45


And equation (2-24) shows that its solution is

y=e if (x)dx fe ff"h 111(x)dx + constant e ff''''

2.4 RICCATI EQUATION

The Riccati equation

dydx

f (x) + g(x)y + h(x)y2 (2-26)

can be thought of as a generalization of the linear first-order equation (2-25).This equation is sometimes called the generalized Riccati equation sinceRiccati actually studied the special case

dyd.t+ by2 = cxm

where b and c are constants.If a particular 'solution yi of equation (2-26) is known, the general solu-

tion y of this equation is

1

where U is the general solution of the linear equation

dxdU + (g + 2y1h)U + h --= 0

which was solved in section 2.3. This follows from the relation

dydx

f gy hy2 =dy,dx

f gyi hy2

1(-dU + gU + h + 2yihU)U2 dx

1 (g + 2yili)U+ hiU2

[ d

dxU

=-- 0

46


Even if a particular solution is not known, it is still possible to reducethe problem of solving equation (2-26) to the problem of solving a linear equa-tion; but in this case the equation is of second order. In order to accomplishthis put

U = e jyh(x)dx

Then U' /U = yh and

Us' = yhU' y' hU yh' U = (y2h2 y' h)U yh' U

( f + gy) HP yh' U

= f hU y (gh + h'1U

1111=fhU+(g+7)(1 ,

Hence, U satisfies the second-order linear homogeneous differential equation

h'U" +

hU' + fhU = 0

The converse of this statement is also true. That is, every second-order linearhomogeneous equation can be transformed into a Riccati equation.

Since miear second-order equations will be discussed extensively inchapter 6, we shall not consider the Riccati equation further here.

47/91

CHAPTER 3

Systems of Equations

In this chapter, a number of important properties of systems of differentialequations are discussed. We shall also show how the solutions of these systemscan be used to obtain solutions to certain types of first-order partial differentialequations.

3.1. FIRST-ORDER NORMAL SYSTEMS

In certain applications it is necessary to deal with systems of simultaneousdifferential equations. The set of n first-order normal differential equations inthe n dependent variables yl, y2, . , yn

dyedx

dyedx

x, Y2, Yrt)

=G2(x, Yl, Y2, Yn)

dyndx

Gn(x, yi, Y.2,9

)

(3-1)

is called a normal system. It provides a standard form to which all normaldifferential equations and all systems of normal differential equations can bereduced. For example, by introducing the new dependent variables yi defined

49

DIIVERENTIAL EQUATIONS

by

dxjfor i= 1, 2, . . . n-1

the nth-order normal equation (1-19) can be transformed into a first-ordernormal system

dy1dx 2

dy2dx y3

dyn -1-Yndx

dYn ,= kykx, yi, Y2, , Yn)dx

However, it is by no means always possible to transform a first-order normalsystem into a single nth-order equation.

In order to obtain a geometric interpretation of the system (3-1) and itssolutions, it is again convenient to think of the variables x, yl, y2, . . yn asbeing the coordinates of an (n 1)-dimensional space. The definitions of adomain and of a neighborhood in this space were given in section 1.1. Based onthese ideas, the fundamental theorem for the first-order normal system can bestated as follows:

Suppose that the functions Gi for 1 , 2, . . . , n are defined and continuousin some domain D and that the partial derivatives

aGt

ayifor 1 i n; 1 n

are continuous in D. Then for each point (x0, y?, A, .(3-1) has precisely one solution

50

. y?) of D equation

SYSTEMS OF EQUATIONS

Y1 = f i(x)y2= f2 (x)

(3-2)

Yn=.fn(x)

which is defined on some interval a < x < p containing the point xo and satisfiesthe initial conditions

fi (x0) = y?, f2 (x0) = )1, fn (xo) =31 (3-3)

The solutions of the system (3-1) can be visualized as curves in the(n + 1)-dimensional space. Then the fundamental theorem states that there isprecisely one such curve passing through each point of the domain D in thisspace.

If H(x, yi, y2, . . yn) is a nonconstant function such that every solutionto the system (3-1) satisfies an equation of the form27

H(x, yl, Y2s ., yn) = constant (3-4)

(the constant may be different for different solutions), then H(x, yl, y2 . yn)is called an integral (or first integral) of the system (3-1). Thus, an integral ofthe system (3-1) is a function H which is not identically constant but is constantalong each solution curve of the system. Therefore, upon differentiating Halong any solution curve of the system (3-1) we obtain

dH aH aH dyi , aH dyndx ax ay, dx -r ayn dx

all aH 0Hax + a + . . . + --ayn Gn= 0

27 Notice that this definition of an integral is consistent with the one given in chapter 1 for the nth-order normalequation when the latter equation is transformed, by the procedure just described, into a firstorder normal system.

.1.51


provided that H possesses continuous first partial derivatives. But each pointof D lies on a solution curve of the system (3-1). Hence, H satisfies the equation

aH aH aH aH+ G1+ G2+ . .+ Gn=0ayi ay2

(3-5)

at every point of D. Conversely, if H is any solution to equation (3-5) in D, itis certainly constant along every solution carve of the system (3-1). Hence, Hmust be an integral of this system. We have now shown that a nonconstantfunction H with continuous partial derivatives is an integral of the system(3-1) in D if, and only if, it satisfies the first-order linear partial differential'equation (3-5).

Each integral H of the system (3-1) determines a family of n-dimensionalhypersurfaces in the region D, and every solution curve lies on one of thesehypersurfaces. Now suppose that we have found n integrals, say H1,-1/2,. . Hn, of the system (3-1) which are defined in the region D. If the Jacobiandeterminant

a (H1, . .

(3(yi, . 9 Yn)

all, 0H1

ay1 ayn

aH, aHnayi aYn

(3-6)

of these integrals is different from zero in D, it will be possible28 to solve the nequations

Hi(x,yi, , Y71)=-- c, , ,11n(X, , Yn)=Cn (3-7)

for yi , . . ., yn as functions of x. And these functions will depend on the narbitrary constants c1, . . cn. The n equations (3-7) therefore implicitlydetermine an n-parameter family of curves in the (n + 1) -dimensional space.

28 This is a consequence of the implicit function theorem for systems of equations. See, for example, ref. 6.

52

HA(x, Yl, Y2) - C

Y2lily, Yl, y2) -C2

Hi(x, Yl, y2) - Cl

Yl

AX

H2(x, Yl, Y2)


H2(x, Yl, Y2) -05fl(x)

Y2 - f2(x)

FIGURE 3-1. Relation between integral curves and surfaces of ordinary differential equations.

But these curves must lie along the various mutual intersections of n hyper-surfaces

H1= constant, H2 constant, Hu = constant

This is shown schematically in figure 3-1 for the case where n = 2. However,since every solution curve of the system (3-1) must simultaneously lie on nhypersurfaces, each of which corresponds to one of the n integrals H1, . .

lin, the solution curves must also lie along these intersections. Therefore,every solution to the system (3-1) can be obtained as an implicit solution of then equations (3-7). That is, the n integrals H1, . . which have a non-vanishing Jacobian in D are just sufficient to completely determine all the solu-tions of the systems (3-1) in D. However, the requirement that the Jacobian(3-6) be different from zero in D implies that the integrals are functionallyindependent 29 in D. We therefore anticipate that the system (3-1) will possessn functionally independent integrals. This turns out to be the case, at leastwhen the region D is chosen to be sufficiently small (see ref. 7).

"The functions HI, . . H,, are functionally dependent in D if there exists a nonconstant function w of HI, . .

H,, such that co(HI, . . =0 for all x, , y,, in D. See ref. 6.

53


For example, consider the normal system

dy,dx

yi, y2) = xy,

dy2 XY2= (.72 (x, Y2) = 2 2dx Y2 Y1

which is defined in some region D which does not include the planes yi = y2.Since the two functions Hi (x, y,, Y2) =y1y2 and H2 = y, + 31x2 satisfy theequations

and

ail,G1+ 3H,

dx ay, aye = 0 + r2 1 xY' + yi xY2 = 031 4-4

aH2 aH2+ Gi+ G2 = 2x + 2yi 2 Y1 +2y2 xY2 = 0ax ay, ay2 A )1-4

we know that H1 and H2 are two integrals of this system. Since

a(H H2) Y2 =2(4-4)a (yi, Y2) 2Yi 2Y2

the Jacobian of these functions does not vanish in D. And this shows that 1/1and I/2 are functionally independent. The solution curves lie on the inter-sections of the surfaces H1= constant with the surfaces H2 = constant. Thus,every solution curve lies on an intersection of a hyperbolic cylinder and ahyperboloid (or cone).

Now let HI, H2, . . Hn+, be any n+ 1 integrals of the system (3-1) insome domain D. Since each of these integrals must satisfy equation (3-5), itfollows that

54


all, all, all,+ G1+ . + G,,--= 0

all2 aH2

ayiax

all2+ = G1+ . Gn= 0

alli,÷i+aHn+1 G1+ . . . + Gn=0ax ayi ayn

And it must be possible to solve these equations for 1, GI, C21 ., Gn atevery point of D. However, this can happen only if the determinant of thecoefficients

a(H1,H2, )

(-% Yn)

all, all, ail,ax ay, ayn

aH2 aH2 aH2

ax ayi ayn

alln+, win+,ax ay, ayn

is zero at every point of D. But this means that H1, . , 1-171+1 are functionallydependent. Hence, every set of n+ 1 integrals of the system (3-1) is function-ally dependent. We have,' therefore, shown that the system (3-1) possessesprecisely n functionally independent integrals (at least in a sufficiently smallregion D).

If n functions /1(x, y,, ., yn), , . . yn) not all identi-cally zero can be found such that

55


is equal to dFldx for some nonconstant function F(x, yl, ., yn), then Fwill be constant along every solution curve of the system (3-1). Hence, Fwill be an integral of the system (3-1). The functions 11, . In can bethought of as integrating factors.3° Although it is usually difficult to find inte-grating factors, they can sometimes be guessed at from the symmetry of theproblem.

For example, consider the system

dy1dx= Y2 Y3

ck.-2

dx Yi

3T-c =Y1 Y2

Then choosing 11= /2 =13= 1 gives

(3-8)

dx(y2 y3)] + 1 x [P (Y3 )] + 1 x (Yi Y2)]

clyi_Lckl+cly2= d(n±y2±y3)dx dx dx dx

Hence, H1= yl +y2+ y3 is an integral of the system (3-8). Now choose2yi, 12=2y2, and 13=2y3. Then

2Yirt (Y2 Y3) + 2Y2[ cick (Y3 Y')] 2Y3[ck (Y1 Y2)]

Lip ddy: ddy:

dx

Hence, H2----yl+A+31 is also an integral of the system (3-8). It is easy toverify that H1 and 112 are functionally independent.

30 See section 2.2.

56


Instead of interpreting the solution (3-2) of the normal system (3-1)as the explicit equation of a curve in an (n+ 1)-dimensional space withcoordinates x, y,, . . yn, we can interpret it as the parametric equation(with parametric variable x) of a curve in an n-dimensional space with coordi-nates y,, . . yn. The parametric variable x may be the arc length or a similarparameter for the solution curves (such as the time in a physical problem).The fundamental theorem can still be used to show that there is a solution curvepassing through every point of any region of this n-dimensional space, providedthe functions Gi satisfy the appropriate smoothness requirements. However,except in a certain important special case (which is the topic of the next sec-tion), it will usually not be true that only a single curve passes through eachpoint of this region.

The simplicity of the first-order normal systems becomes apparent wheniequation (3-1) is written in vector notation. To this end we notice that thecoordinates of a point yi, . . yn in the n-dimensional space can be interpretedas the components of an n-dimensional vector y and we write 31

y= (31, y2, - yn)

A vector is said to be a function of a single real variable x if its componentsare functions of x. Thus, if (as in the case of the solution curve (3-2) of thesystem (3-1)),

(x), . yn=fn(x)

we write

Y= f (.0 (x), , fn(x))

31 Addition of two vectors and multiplication of a vector by a scalar are defined componentwise. Thus, if y=(y,, y.) and x= (x,, . x.) are vectors and a is a number, y x= ((y, . , (Y.+ z,)), ay= (ay,,ay.). The length or magnitude of the vector y is denoted by iYi and is defined as the distance from the origin to thepoint y,, . . y. Thus, iYi = (Y1+ . . . +yf,)'/2. Notice that the magnitude of a vector is zero if, and only if, itscomponents are all zeros. The dot product or inner product of two vectors y and x is denoted by y x and is definedby y x=y,x, + . . . + yx. The dot product satisfies the Schwarz inequality 1Y xi '4 iYi xi, where the verticallines on the left are the absolute value of a pure number and those on the right denote the magnitude of vectors. Thus,if fyl s 0 and lx/ # 0 then IY xi/iYi ixl = 1 and this quantity can be interpreted as the cosine of the angle betweenthe vectors y and x. If y x=0, we say that y and x are orthogonal; and if ly xIlly1 lx1=1, we say that the vectorsare in the same direction. The vector (1, 0, 0, . . 0) has unit magnitude and is in the direction of the y,-coordinateaxis. The vector (0, 1, 0, 0, . . 0),has unit magnitude and is in the direction of the yrcoordinate axis, etc. We denotethese vectors by ki, k2, etc. For more information, see ref. 8.

57


Differentiation of vectors is defined component-wise, and we write

41= cly,i)_df(x) ( dfi(x) dfn(x)\dx dx dx dx ' ' dx )

If f(yi, . . yn) is a function of the n variables . . yn and y=. yn) denotes the vector corresponding to these variables, we write

Pri, , yn) =f(y)

and say that f is a function of the vector variable y. More generally, we say thata vector is a function of a vector variable if its components are. Thus, if yi=fi(x), . . , Yn=fn (x) , then

y=--- f(x)= (fi(x), . . .,fn(x))

is a vector function of the vector variable x.Notice that there is no such thing as the derivative of a function of a vector

variable with respect to that variable, but only partial derivatives with respectto the components of the vector variable. Thus, if ki, . . kn denote the unitvectors in the directions of the coordinate axes, we define the vector gradientoperator V by

Vfix) = k, afa(xx)+ . . .± kn afix)

Just as in the three-dimensional case, the effect of operating with the gradientoperator on a scalar produces a vector.

A vector function is said to be continuous if its components are continu-ous. Thus, in the system (3-1), each Gi is a function of a real variable x and avector variable y and we write

Gi(x, Yi, yn) = Gi(x, Y)

If we let G be the vector whose components are then G is a vector functionof the real variable x and the vector variable y and

58


G(x, y) = (Gi(x, Y), , Gn(x, Y) )

Hence, upon using the component-wise definition of differentiation of vec-tors, the first-order normal system (3-1) can be written in the concise form

G(x, y)dx

When the first-order system (3-1) is written in this form, the analogy betweenthis system and a single first-order equation becomes particularly apparent.

3.2 AUTONOMOUS SYSTEMS

When the functions Gi in the system (3-1) do not involve the variable xexplicitly, the system is said to be autonomous and it can be written in theform

dxdyi

-- eTi(yi, y2, ,,yn)dy2dx y29 5Yn)

dyn =dx

Lyn (Y1, Y2,

(3-9)

If the independent variable x is thought of as representing time, autonomoussystems can be interpreted as time-independent, or stationary, systems.

The first-order normal system of n 1 equation

dy1dxGi(Yi! Y29 , Yni5x)

dyn-i.- .

dz= (. 72ki, Y2, , YnI, x)

urtikYl, Y2, , Yn-1, x)

dy2

dx

488-942 0- 73 - 5

(3-10)

59


can always be transformed into an autonomous system in n variables which hasthe property that at least one of the functions on the right side never vanishes.In order to do this, it is only necessary to introduce a new dependent variableYn by

Yn

and then rewrite the system (3-10) in the form

dyidx= Gi (yi, y2, , Yu )

dY2=G2(Yi, Y2,dx Yn)

dyn-t= Y2, yn)dx

dyndx.

Conversely, if one of the Gi's of the autonomous system (3-9) in n variablesdoes not vanish in some domain Do, it can be transformed into a nonautonomoussystem in n-1 variables plus an additional equation which can be solved todetermine the variable x once the remaining variables have been found. Wemay assume without loss of generality that the notation has been so chosenthat Gn does not vanish in Do. Now in order to accomplish this transformationput

and

60

Yn (3-11)

Gifor 1 -a-5. i s (n 1) (3-12)


Then the system (3-9) becomes

dy,7----cn(Yi 9. , Yn-b x

dx

dy2=-4,2tY19 ., yn-19

(3-13)

dy_, ,n-1 (Y11 ., Yn-19 x

dx

and the equation which determines x once the system (3-13) is solved is

dx 1

dx Gn (yi , Yn-1, X)(3-14)

It can now be shown that every solution to the system (3-13) in Do is asolution of the system (3-9), and conversely. Thus, we may say that the systems(3-9) and (3-13) are equivalent in Do.

The solutions of the system (3-9) in n variables can be found by solvingthe n 1 simultaneous equations (3-13) to determine yl, . . in-1 as functionsof yn. And then a single equation obtained by substituting these solutions intoequation (3-14) can be solved to determine the parametric variable x as afunction of y,,. In order to emphasize the connection between the autonomoussystem of n equations (3-9) and the system of n 1 equations (3-13), theformer system is sometimes denoted symbolically by

dy, dy2_ dyn dx---Gi G2 Gn

or, when finding the variable x is not important, by

dy, dy2_ dynGi G2 Gn

61


Thus, for example, consider the autonomous system of three equations

dxdy1

(yi, Y2, Y3) myiy2A

dy2dx

=G2 (yi, Y2, y3) nYIYI

dY3

dx G3 (y1, Y21 Y3) = Y1Y2Y3

Upon putting i= y3 we get

dyi = (yi, y2, i) = myii

CIY2' C2 (Y1 Y2 i)ny2

di

and x is determined by the equation

dx 1

di C3 Y1Y2i

The system (3-16) has the solution

mx2I2 my2 /2cie =-- cie

C2

Y2 "71X

Finally, equation (3-17) shows that x is related to x by

x= c3+1 expCC c2 2

dx

62

(3-15)

(3-16)

(3-17)


The interpretation of the solution of a normal system of n equations as acurve in an n-dimensional space is particularly useful when the system is autono-mous. This is the case because for autonomous systems, unlike nonautonomoussystems in general, there is usually only a single curve passing through eachpoint in this space. We can show that this is the case by applying the funda-mental theorem given in section 3.1 to the transformed system (3-13). Thistheorem applies to the y1, . . space in which the 'system (3-13) isdefined. But in view of equation (3-11), we see that this is the same as the.n-dimensional y1, . . *y space in which the untransformed autonomoussystem (3-9) is defined. Thus, suppose that the functions Gi have continuousfirst partial derivatives with respect to all their arguments32 at every pointin some domain D of the n-dimensional y1, . . yn space and that there isno point of D where all the Gi's simultaneously vanish. Then, because of thecontinuity of the Gi's, we can assert that for any point y?, . . yg of D thereis a neighborhood Do in which at least one of the Gi's is never equal to zero.Within this neighborhood we can transform the normal system (3-9) into thesystem (3-13) and apply the fundamental theorem in the region Do. Since theGi's have continuous partial derivatives, we can conclude that the system (3-9)has precisely one solution curve in the n-dimensional yl, . . yn space pass-ing through the point 4, . . . , 31. Now suppose the notation is chosen so thatG. does not vanish in D.Q. Then G. does not change sign in Do, and equation(3-14) shows that x is a monotonic function of x along any solution curvein Do. Hence, we can take x as the parametric variable for the solution curveinstead of x. Since this is true for every point of D, we arrive at the follow-ing conclusion:33 If the function G which appears in the first-order autonomoussystem

dy= G(y)dx (3-18)

possesses. continuous partial derivatives at every point of some domain D, andif IG(y)I does not vanish at any point of D, the system (3-18) has exactlyone solution curve

32Notice that in this case the existence of the partial derivatives guarantees the continuity of the functions G.33We are using the vector notation introduced in section 3.1.

63


y = f(x, yo) (3-19)

passing through each point yo of D.Thus, we can think of the "solution vector" (3-19) as tracing out a con-

tinuous curve in the n-dimensional yt, . . yn space, which passes throughthe point yo when the parametric variable takes on some value, say xo. Such"solution curves" are called integral curves of the system (3-18). These ideasare illustrated in figure 3-2 for the case where n= 3.

Notice that in order to ensure that the system (3-18) has, only one integralcurve passing through each point of D, it is necessary to require that IG(y) I,the magnitude of G(y), does not vanish at any point of D. Points where IG (y) I =-0 are called critical points of the autonomous system (3-18).

If yo is a critical point of the system (3-18), this system must possess theconstant solution

y= yo

which is called an equilibrium solution.For example, it may be verified by inspection that the solutions to the

autonomous system

Y2

FIGURE 3-2. Relation between solution vector and integral curve of differential equation.

64

idydx mY1

ligdx nY2

with n > m, are


ciemx

y2 =--- c2enx

The point (0, 0) is a critical point. The integral curves of this system are theloci of points

(y2)1n= k(yi)"

where k= di/a. These curves are shown for the case where m and n arepositive integers in figure 3-3. It can be seen from this figure that there is

FIGURE 3-3.Integral curves for sample problem.

65

Li


exactly one integral curve passing through each point except the critical point(0,0).

A vector function which is defined at each point of an n-dimensionalspace is said to constitute a "vector field." Since a unique vector-valuedfunction G(y) corresponds to each autonomous system (3-18) and conversely,we can say that each first-order autonomous system is characterized byits vector field.

Consider a portion of an integral curve of the system (3-18) which passesthrough a point y which is not a critical point and which lies in a domain Dwhere G(y) is continuously differentiable. Let S denote the arc length meas-ured in some definite direction along this curve. Then

Or

(c/S)2=(dY1)2+ (dy2)2+ . . . + (dYn)2

dS = dy dydx V--dx dx

But equation (3-18) shows that

dxd9--± VG G=7±: iGi

Hence, taking S as the independent variable in equation (3-18) shows that

dy_ dy dx__t_ GdS dx cLS-- IGI

However, the vector dy/dS is the unit tangent vector to the integral curvepassing through y; and since there is an integral curve passing through eachpoint of D, we conclude that the vector field G(y) is tangent to the integralcurves of the system (3-18) , except possibly at the critical points.

When n =3, we can imagine that the variable x represents time and theautonomous system (3-18) represents the steady flow of a fluid in space. Ateach point y in a certain region of this space the vector G(y) describes thevelocity of the fluid at that point in both magnitude and direction. The flow

66


is steady because its velocity depends only on position and does not vary withtime. The integral curve passing through the point yo may then be interpretedas the streamline followed by all the fluid particles which have passed throughthe point yo. The velocity field is tangent to the streamlines at every point ex-cept at the critical points which correspond to "stagnation points" where thefluid velocity vanishes. At these points, it is possible for two or more stream-lines to meet or for a given streamline to abruptly change direction.

By again applying the results obtained in section 3.1 to the autonomoussystem (3-9) by means of the transformed system (3-13), we arrive at thefollowing conclusions: Suppose that D is a region of yl, . . yn space whichcontains no critical points of the system (3-9) and:that G has continuous partialderivatives at every point of D. Then a function H(y) of the vector variabley in D which does not depend explicitly on x is a solution of the first-orderlinear partial differential equation

Gl(y) -aH+ . . .± GtiYayi a rn

(3-20)

if, and only if, it is an integral of the autonomous system (3-9). If, in addition,the region D is sufficiently small, the system (3-9) possesses precisely n 1functionally independent integrals which are independent of x. This shows thatequation (3-20) possesses n 1 functionally independent solutions.

The level surfaces of these integrals are hypersurfaces in the n-dimensional, . . yn space. The intersection of any n 1 of these hypersurfaces (no

two of which correspond to the same functionally independent integral) isan integral curve of the system (3-9) in this space.

The partial differential equation can be written more compactly by usingthe vector notation introduced in section 3.1 to obtain

G v H=-- 0

In this form the equation has an immediate geometrical interpretation. As inthree-dimensional space, the vector VH is perpendicular to the level surfacesof the function H. Hence, the differential equation (3-20) merely states that thevector field ;G is everywhere tangent to the level surfaces of its solutions. Thisis consistent with the facts that the integral curves of the system (3-9) lie onthese level surfaces and that the vector field G is tangent to these integralcurves.

67


3.3 SOLUTIONS OF THE FIRST-ORDER LINEAR PARTIAL DIFFERENTIAL EQUATION

Since equation (3-20) cannot have n functionally independent solutions,there must exist for any n solutions H1, . . lin of equation (3-20) a non-constant function co such that

co(111, H2, . 117,) 0

for all yl, . . yn in some region D. Now it can be shown (ref. 6) that thisequation can always be solved for H, to obtain

lin=F(111, H21 ..,H -1)

provided that H1, . , Hn-1 are functionally independent. But since Ht, ,11, were any solutions of equation (3-20), this shows that every solution ofequation (3-20) can be expressed as a function of any n -1 functionally inde-pendent solutions. Conversely, it is easy to verify by direct substitution that ifHI, H2, . Hn-1 are any solutions of equation (3-20) and F is any continu-ously differentiable function of H1, . . Hn_1, then F(1/1, H2, . Hn-1) isalso a solution of equation (3-20). Thus, we have shown that H is a solution ofthe linear partial differential equation (3-20)ff, and only if, there is a functionF such that

H(y)= F (Hi(y), . . Hn--1(Y))

where H1(y) , . . Hn-1(y) are any n -1 functionally independent integralsof the autonomous system (3-9) which do not depend on x.

Thus, the most general solution of the partial differential equation (3-20)can be found if the system (3-9) of ordinary differential equations can be solved.The differential equations (3-9) are therefore called the characteristic equationsof the partial differential equation (3-20) and their integral curves are called,characteristic curves of the partial differential equation.

For example, the characteristic equations of the partial differentialequation

aH , , , aH , , aH(Y2 y3) .7 -r ky3 y1) -r y2 =0 (3-21)ayi ay2 ay3

68

SYSTEMS CF EQUATIONS

are the equations of the system (3-8). But we have shown that this system hasthe two functionally independent integrals

H1 ----- + Y2 ± Y3

H2 ="-- + ± A

Thus, for every differentiable function F, the function

H =F(H1, 112)

must be a solution of equation (3-21). This can easily be verified by substitut-ing the relations

aH aF 2 aFan aH, --Yi aH2

for 1, 2, 3

into equation (3-21).The level surfaces of the integral H1 are planes, and the level surfaces

of the integral H2 are the surfaces of spheres. The intersections of the surfacesH1= constant with the surfaces H2= constant are therefore circles. Hence,the characteristic curves of equation (3-21) are circles.

3.4 QUASI- LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

There is also a close connection between the quasi-linear partial differentialequation

of ofGi(yi, ) . . . +Gn-i(Yi, ,ayt

= Gn (Y1, Yn-if) (3-22)

and the linear partial differential equation (3-20) and, therefore, also betweenthis equation and the autonomous system (3-9). In order to show this, let

H = CY', . , Yn)

69


be any solution of equation (3-20) such that ali/ayn is not identically zero.34Then the implicit function theorem shows that there is a neighborhood D ofany point yo for which

such that the equation

has a solution

all ooayn

h(yi, ,yn)=

yn= f (yi, , yn-1, c1)

(3-23)

(3 - -24)

(3-25)

in D. This means that when equation (3-25) is substituted into equation (3-24),the resulting expression is identically constant for all values of y1, . , yn-iHence, upon differentiating this expression successively with respect toyi, . . yn _1, we obtain

aH +ail ofv

aY1 aYn 8Y1

aH +all of =0ayn-i ayn ayn-1

But when these relations are substituted into equation (3-20), we find that

aftrn

ayiaYn

af \_aYn-i)

And, in view of equation (3-23), this shows that equation (3-25) is a solutionof equation (3-22).

34 This means thaz II is not independent of yn.

70


Now there are n -1 functionally independent integrals H1, . . Hn_iof the system (3-9) , which are therefore functionally independent solutionsof equation (3-20). If there is some point y at which these integrals all satisfycondition (3-23) , we can find at least n -1 functions

Yn =fi (Yi

Yn= fn-1(y1,

, Yu-1, C1)

, Yn-1, Cn-1)

which, respectively, satisfy the equations

Hi= hi , , Yn) =

Hn-1 = hn-1(Yi, 1- , Yn) = Cni ,

(3-26)

(3-27)

in some neighborhood D of the point y. Therefore, each of the functions (3-26)is a solution to equation (3-22) in D. Since equations (3-26) determine explic-itly the same surfaces as equations (3-27) determine implicitly and sincethese surfaces determine the integral curves of the system (3-9), the equationsof this system are also called the characteristic equations of equation (3-22).

For example, the partial differential equation

(y2.1) -aL.Lv

,

aafari Y" Y2

(3-28)

has the same characteristic equations as the linear equation in the precedingexample, namely equations (3-8). These equations, as we have seen, have thetwo functionally independent integrals

71


H1= yi + y2 + Y3

H2--=A+A+A

And equation (3-28) therefore has the solutions

f i = ci (Yi + Y2)

f2=± N/q ( A + A )

72

CHAPTER 4

Elementary Methods for Second-Order Equations

In this chapter several methods for obtaining solutions to second-orderdifferential equations are presented. Even though each of these methods appliesto a relatively narrow class of equations, they are still of sufficient generalityto be useful in practice. Ho Weyer, unlike the methods presented in chapter 2for solving first-order equations, most of the techniques presented in this chap-ter will not by themselves yield solutions to equations. They are, in fact,methods for transforming certain types of equations into simpler equationswhose solutions can hopefully be found by known methods. We have alreadyfound a number of methods for solving first-order equations. The only othergeneral class of equations for which it is possible to find numerous solutionsis the class of linear equations introduced in chapter 1. Therefore, in section4.1 we present some techniques for reducing second-order equations to equa-tions of the first order. And in section 4.2 we present techniques which trans-form certain types of nonlinear equations into linear equations. Finally, insection 4.3 we present a number of unrelated methods. Although many of themethods presented in this chapter are applicable, with some obviG--,5 modifi-cation, to equations of higher order, we shall, for simplicity, limit the dis-cussion to second-order equations.

As indicated in chapter 1, the most general second-order differentialequation is an equation of the form

F' \ 0dx2' dx' ' x

whereas, the most general normal second-order equation is of the form

(4-1)

73


d2y (dy=G xdx2 dx"

(4-2)

A general solution of equation (4-1) or of equation (4-2), if it exists, willinvolve two arbitrary constants. As in the case of first-order equations, theremay be any number of general solutions to these equations. In addition, equa-tion (4-1) may possess singular solutions. These solutions are discussed insection 1.5. The methods presented in that section should be used to find anysingular solutions which may be present since such solutions can be importantin physical problems.

Finally, the second-order linear equation is an equation of the form

dyP0(x)

d2y+ (x) cw+ p2(x)y= p3(x) (4-3)

We shall assume the coefficients are continuous. Upon dividing through bythe leading coefficient po(x), we obtain the (essentially) normal form

dz dy-L+P(x) d

q(x)y= r(x) (4-4)

This differential equation is equivalent to equation (4-3) in any interval inwhich po(x) does not vanish.

4.1 EQUATIONS WHICH ARE REDUCIBLE TO FIRST-ORDER EQUATIONS

In this section a number of techniques, which can be used to reducecertain types of equations to equations of the first order, are described.

4.1.1 Dependent Variable Missing

When a second-order differential equation does not explidtly contain thedependent variable y, it must be of the form

F(ch 421- =dx2' dx'

But this equation can be written as a first-order differential equation

74

(4-5)

for the quantity p defined by

ELEMENTARY METHODS FOR SECOND-ORDER EQUATIONS

F P, x) = 0

dy (4-6)

It may or may not be possible to solve this first-order equation in any givencase. However, suppose that it can be solved and that its integral is

f(x, p) = constant = C1 (4-7)

At this stage, there are two possible ways of proceeding, depending on whetherit is easier to find an explicit formula for p as a function of x which solves35equation (4-7) or whether it is easier to .find an explicit formula for x as afunction of p.

First, suppose that the former case occurs. Thus, we can find an explicitformula

p=g(x,C1)

for a solution to equation (4-7). Then, in view of equation (4-6), this equationcan be immediately integrated to obtain a general solution

y= ex, Ci)dx+C2

of equation (4-5).Next suppose that it is easier to find an explicit formula

x-----h(P, CO (4-8)

for a solution to equation (4-7). Substituting equation (4-8) into the relation

_dydpP dpsdx

'See section 1.1.

488 -942 0 - 73 - 6

75


shows that

Hence,

dy dx dhdp P dp P dp

y=fp dh+C2

And, upon integrating this by parts, we obtain the equation

y= ph (p, (4-9)

which together with equation (4-8) determines the solution y of equation (4-5)parametrically as a function of x (with p being the parametric variable). Thisparametric function involves two arbitrary constants.

For example, the equation of the curve followed in the pursuit of a preywhich moves along the y-axis (ref. 9) is

K V1+ y = (a- x)Y"

where K is the ratio of the velocity of the prey to the velocity of the pursuer.Put y' =p to obtain the first-order separable equation

K 1( 1- F p 2 (ax) ddx

which can easily be solved (section 2.3) to obtain

dy 1 1

dx= [ci (a x)K u; (a x)-K

If K 1, this equation beccim:ts, upon integration,

r-1-

1(a x)1--KY= 2 L1K "2 C1(K-1)

(If K=--- 1, the integration yields a logarithmic term.)

76


4.1.2 Independent Variable Missing

The general form of the second-order equation in which the independentvariable does not appear explicitly is

We again define p by

td2y dydx2'

Y)

dydx

Then substituting this together with the relation

dx2 dx ,dy dx dy

(4-10)

(4-11)

(4-12)

into equation (4-10) shows that p is determined by the first-order differentialequation

F(2 p, p, y)=0

Suppose that this equation can be solved and its integral is

f(p, y) = CI = constant (4-13)

It is again possible to proceed in two different ways. First, suppose that it iseasier to find a formula for the solution p= ey, C1) of equation (4-13) for pas a function of y. Then combining this with equation (4-11) and integratingprovides a general solution

fg(y, 2-_ x

of equation (4-10).On the other hand, if it is easier to find a formula

y==h(p, CI) (4-14)

77


for the solution of equation (4-13) which expresses y as a function of p, equation(4-12) shows that

dx _1 cly_l dh'dp p dp- p dp

But this equation can be integrated by parts to obtain the equation

1x=C2+-1 h(p, + h(p,Ci)dp (4-15)

which, together with equation (4-14), determines the solution to equation (4-10)parametrically.

For example, the differential equation

d2y (dy)y v

dx2 dx=0 (4-16)

occurs in the field of fluid dynamics.36 Substituting equations (4-11) and (4-12)into this equation yields the first-order equation

ypdp + vp2 = 0dy

Since this equation is separable, it can be integrated to obtain p= Cilyv.Hence, combining this with equation (4-11) and integrating shows that

ifv+

1CIX + C2

11+

is a solution of equation (4-16).Notice that, when A, B, C, and D are constants, the differential equation

y"=A+By+Cy2+Dy3

36 More specifically, it governs the, velocity field in a boundary layer in the neighborhood of the separation point.

78


does not explicitly contain, the dependent variable x. When the procedurejust described is applied to this equation, we encounter integrals which usuallycannot be evaluated in terms of elementary functions. These integrals, however,can always be expressed in terms of elliptic functions. The differential equationis therefore called the elliptic equation. For more details concerning the proce-dures involved in evaluating these integrals and the properties of ellipticfunctions, the reader is referred to reference 10 (chapter 6 and section 11 ofchapter 7).

Finally, notice that a linear homogeneous equation does not contain theindependent variable explicitly if, and only if, it has constant coefficients."

4.1.3 Homogeneous Equations

The gt.,. eral definition of a homogeneous function of degree k has beenOven in section 2.3. We shall now show that if the second-order differentialequation (4-1) has certain homogeneity properties, it can be solved by ele-mentary means.

4.1.3.1 Equations homogeneous in the dependent variable and its deriva-tives.- First, suppose that equation (4--1) is homogeneous of degree k in thevariables y, y', and y"; that is, it satisfies the homogeneity condition

F (ty" , ty' , ty, x) = th'F (y" , y' , y, x) (4-17)

for any number t. Hence, if we put t= 1/y, equation (4-17) becomes

FIY Y Y

x)= F (y" , y' , y, x)

which shows that the differential equation (4-1) can be written in the form

FY Y

1, = 0 (4-18)

by factoring out y". In order to reduce the order of this equation, define thefunction u(x) by

37 We assume that the reader knows how to solve linear equations with constant coefficients.

79


Then

=U + U2

(4-19)

Upon substituting this, together with equation (4-19), into equation (4-18)

we obtain the first-order equation

F (u' + u2 , u,1, x) =0

Suppose this equation c be solved and its integral is

f(u, x) =C

Then solve equation (4-21) for u as a function of x to obtain

u =g(x, C1)

(4-20)

(4-21)

(4-22)

But substituting this into equation (4-19) and integrating shows that theoriginal equation (4-1) has the solution

In y= (g(x, Ci)dx+C2 (4-23)

If the differential equation (4.1) is linear, the first-order equation (4-20)

obtained by this ethod will be a Riccati equation which will usually bemore difficult to so e than the original equation.

As an example f the method, consider the equation

F(y", y', y, x) = yr" Y2+ Y2x =0 (4-24)

and replace y, y', and y" by ty, ty', and ty ", respectively, to get

(ty) (ty") (ty')2 +

80

(tY)2x= t2 (17" 3/2 + Y2x) = t2F (Y" Y', y, x)


This shows that F is homogeneous of degree 2 in y and its derivative. Hence,upon introducing the transformation (4-19), the differential equation (4-24)becomes

u'+u2-u2+x=te +x=0

But this equation can be immediately integrated to obtain

x2u+-2 =Ci

Hence,

In y= f (CI -c)dx+ C2

The solution to equation (4-24) is, therefore, given by

y= C3 exp (Cix

Triere we have put C3 = eC'2.4.1.3.2 Isobaric equation. -The isobaric equation of the first order in

normal form was encountered in section 2.3. In the general case, the function

F dmy dm-ly dyVben' ' dx' Y' xj

is said to be isobaric if there exist numbers k and I such that

dyF(tl-m c17---7-LY t1 -m÷1 dm-1Y

t1-1,dxdxm dxm-1' tly,

tkF(dmY dm lYdy

)dxm'

(4-25

for all values of t. And the differential equation

Fb(m), Pm-1), , r', r, x] = 0

81


is said to be isobaric if the function F is isobaric. It is easy to see that the dif-ferential equation (2-17) is of this type (for m=1). For the second-order dif-ferential equation

F (y" , y', y, x) (4-26)

the condition (4-25) becomes

F (t1-2y" , tl-ly' , tly, tx) = tkF (y" , y', y, x) (4-27)

Thus, in particular, upon replacing t by 1/x, equation (4-27) becomes

A,-f7, 1) =iix F 61" y', y, 9 =0

And this shows that equation (4-26) can be written in the form

F(y_21 xi511, y, 1)=0 (4-28)

However, when we introduce the new variables

-1u x y (4-29)

and

e= ln x (4-30)

into this equation, we obtain an equation

Fr12+ (2/-1) dd+/(/__s1) u, 1]-= (4-31)

in which the independent variable e does not appear explicitly. Since thisequation can always be reduced to a first-order equation by the methods ofsection 4.1.2, it follows that the isobaric equation of the second order can alsobe reduced to such an equation.

82

ELEMENTARY METHODS F-OR SECOND-ORDER EQUATIONS

The homogeneous linear isobaric equation of the second order is an equa-tion of the form

x2y" --1-poxy' qoy== 0 (4-32)

where po and qo are constants. It is known r.s the homogeneous Euler equation.Since this equation satisfies condition (4-27) with 1= k= 0, the change ofvariables u= y, e= in x will transform it into the equation

d2u dude2+ (P0-1) --de+ qou= °

The independent variable does not appear explicitly in thi: equation and thecoeffic:nts are constants.38

This equation, therefore, has two solutions of the form u= ece, one for eachof the roots of the equation.

C2+ (po- 1)C +qo =O

It is easy to see that the general Euler equation

x2y" +pox?? + q0,=r(x)

can also be transformed into an equation with constant coefficients by thechange of variables

u= y e =-- ln x

A less elementary exampl an isobaric equation is provided by theequation

3 \ 2F (Y" x)=YY" 4 r' +yL) =0 (4-33)

It is easy to verify that, when the function F is given by equation (4-33), thecondition (4-27) will be satisfied for any value of 1 provided that k=21-2.Hence, upon introducing the change of variables

"Which is as it should be since the equation is linear.

83


Yx

e=lnxi

the differential equation (4-33) is transformed into the equation

..d24 liclL__3( du \2±(1-1u2 =0u.de2m2 ck. 4 \ ck) \ 2

(4-34)

(4-35)

Since in this case any value of l can be used, we choose its value tosimplify equation (4-35). This is accomplished by setting 1= 2, whereuponequation (4-35) becomes

d24 ( du \2_u de27i de)

But this is a special case of equation (4-16) with v = 3/4 and, therefore, hasthe solution

4014.cie+ C2

Hence, byusingequations (4-34) and (4-35), the solution to equation (4-33)is found to be

y=x2( Cl In x+C2\ 4

4.1.4 Method of Variation of Parameters

This method applies only to linear equations. We have seen in chapter 1that the homogeneous equation (associated with eq. (4-4))

y" +p(x)yi + q(x)y= 0 (4-36)

possesses exactly two linearly independent solutions in any interval in which itscoefficients are continuous, and that the general solution of the nonhomoge-

84


neous equation (4-4) in this interval is the sum of any particular solution of thatequation plus an arbitrary linear combination of two linearly independentsolutions of (4-36).

We now suppose that by some means a nontrivial solution yH of the asso-ciated homogeneous equation (4-36) has been found. Substituting

vyff

(where v is a function to be determined subsequently) into equation (4-4) andusing the fact that yH satisfies the homogeneous equation (4-36) CD simplifythe result shows that

'riff)" (2Y;i+PYR)v' =r

But this is a first-order linear equation for v'. It can therefore be solved by themethods of section 2.3, to obtain

yHv elPdx =-- r(x)ykx)ef p(=» dX Cl

When this equation is solved for v' and integrated, we obtain an expression forv (in terms of known functions) which contains two arbitrary constants. Hence,substituting this expression into y=yHv gives the general solution to equation(4-4).

For example, it is easy to find by inspection that yH= x is a homogeneoussolution of the equation

x3+ 3x2x 23,14-x2 1 -Ex2

y- 1-1-x2(4-37)

Hence, substituting y= xv into this equation shows that v' satisfies the first-order linear equation

2 , =x2+3v"+x(x2+1)

vx2 +1

which can easi be integrated to obtain

85


X2V' X3

x2+1 x2+1 ±C1

But solving this equation for v' and integrating shows that v is given by

1v=--x2

(i+C1 x--x)+C2

Therefore, the general solution to equation (4-37) is

x3y= xv=-2+ CI (x2 1) + C2x

4.1.5 Equations Invariant Under a Transformation Group

The group-theory method for reducing the order of equations is a generali-zation of all the techniques previously discussed in this section. However, thistechnique cannot be applied routinely since, as will be seen, it is necessary tofind a group under which a given equation is invariant and there is no con-structive procedure for accomplishing this. The method is more useful forworking backwards to find general methods for solving particular classes ofequations by starting with a given group and testing equations for invarianceunder this group in much the same way that equations are tested to see if theyare homogeneous.

A single parameter Lie group" in two dimensions is a family of coordinatetransformations

xi= f(x, y; a)

yi =g(x, y; a)(4-38)

i- which the members of the family are individuated by the values of the param-eter a. In addition, the family must contain an identity transformation which,without loss of generality, we can associate with the value a= 0. Thus

"Various continuity hypotheses, which will not be presented here, are required for a rigorous treatment of thetheory.

86


x = f(x, y; 0)

y=g(x,y; 0)

Consider an infinitely differentiable but otherwise arbitrary function of thecoordinates F(x, y). If the point (x, y) is transformed by the group (4-38) intothe point (x1, yi) for a particular value of a, the value of F at (xi, yi) is givenby the Taylor series whose leading terms are

F(xi, yi) =F(x,y)+a(-a81 2-Fci ag a F(x, .mx aay c,)=0

=F(x,y)+a[e(x,y)4 71(x, y) 41F(x,y)+. .

(4-39)where we have put

e(x, (1).=0 and 71(x, y)ace ),0 (4-40)

In particular, if F is taken successively as x and y, then

xi=x+af(x, y)+. .

ri=Y+071(x,y)+ .

(4-41)

When a is infinitesimally small, the transformation (4-41) differs only infin-itesimally from the identity transformations; therefore, e and 7) are referred toas the infinitesimal transformations of the group. The operator

ecx,a

7)(x, y)a

is called the infinitesimal operator of the group. By using this operator notation,the Taylor series (4-39) can be expressed as

F (xi, y,) = F(x, y)-F aUF cx2U2F .

87


But this series can be formally summed to obtain

F(xi, yi) = eauF(x, y)

In the particular cases where F is taken to be the coordinates themselves, thisbecomes

= eaux

yi = eauy

These equations are equivalent to the equations (4-38) which define the originalfinite transformation. This shows that the finite transformation is completelydetermined by the infinitesimal transformations.

In order to illustrate these ideas, consider the magnification group

Yi=g(x, y; a) =--- eay(4-42)

It is apparent that we obtain the identity transformation when a = 0. Insertingthe equations of this group into equations (4-40) shows that the infinitesimaltransformations of this group are

--( X)a=o

ag77(x, r)=(aa)

and therefore that the infinitesimal operator is

a a--ryu=x

ax ay

Applying this operator repeatedly to the coordinates yields

88

Hence,


Ux=x Uy= yU2x=x U2y= yU3x= x U3y= y

xi=x+ ax-F-2

cr2x-F-1a3x--1- . .

3!

rii=y+ ceY+ ct2Y+-3013Y+

And upon summing these series, we obtain the original group

xi= eax yx =eay

The differential equation

G(x, y, y , y") = 0 (4-43)

is said to be invariant under the group (4-38) if introducing the new variablesxi and given by equations (4-38), into this equation leads to the equation

G(xi, yl, yi, y;') =0

This means that the change of variable given in equations (4-38) does not alterthe form of the differential equation (4-43).


G(x, y, y, y") .=.xy" F( -, y')= 0 (4-44)

is invariant under the magnification group (4-42) since it follows from equations(4-42) that

89


y =yi dy dyi d2y d2yi--x xi' dx

x' dx 2

= xidX2

We state without proof that a necessary conditiori for the differentialequation (4-43) to be invariant under the group (4-38) is that

UoG= 0 (4-45)

where the operator Uo is defined by

a a au07- erx+71-6;+q7+x-ay

with

and

dn , dTx Y cl;

d 2 7 d e d26x dx2Y,

,

dx dx2In performing the partial derivatives we treat x, y, y', and y" as independentvar:ables; whereas, in performing the total differentiations we treat thesequantities as functions of x.

For example, since we have shown that

e= x 7 1 = Y

for the magnification group (4-42), it follows that

dxC=

,

T=

dxyl I

90

y' 0=0


for this group. The operator Uo is, therefore, given by

But since

Ue= xa + y -a- (4-46)ax ay

(7- , = F ,ax x x

aay x

applying the operator (4-46) to equation (4-44) shows that

aFUoG=-yay -FyaF =0ay

which is consistent with the result obtained in the previous example.We have shown in chapter 3 that the system"

dx dy dpe(x,y) 71(x,Y) 4(x,Y,P)

(4-47)

has two functionally independent integrals, u(x,y) and v(x, y,p), one of whichcan be chosen independent of p. The following result allows us to apply theideas of group theory to reduce the order of a differential equation: If thedifferential equation (4-43) is invariant under the group (4-38), then theequation obtained by introducing the integrals u and v as new variables in thisequation is nf the first order.

Thus, v have seen in the preceding examples that equation (4-44) isinvariant under the magnification group (4-42) and that, for this group,

e=x 71Y 4=0 X=0

Hence, the system (4-47) becomes

dx dy dpx y 0

40We have written p = y' .

488-942 0 - 73 - 7

91


which has the two functionally independent integrals

But since

we see that

u= 2` andx

dp dv du dv p ydx dx du dx du (x -x2

vxy

duu)

d

Hence, upon introducing the new variables u and v into equation (4-44), weget the first-order equation

dv F (u, v)du u v

It is easy to see that equation (4-44) is the general isobaric equationwith 1=1. For this case the group-theory method is the same as the methodgiven in section 4.1.3.2. In fact, all the methods of solution given up to nowin this chapter are equivalent to the group-theory method when used in con-junction with certain well-known groups.

For example, thA equation with the dependent variable missing, treatedin section 4.1.1, is invariant under the translation group

x1=---ac

yi=y+a

And the equation with independent variable missing, treated in section 4.L2,ie invariant under the translation group

x1= x ay1=-3'

92


The isobaric equation with 1 =0, treated in section 4.1.3.2, is invariant underthe affine group

xl = eax

Y1 Y

And the homogeneous equation treated in secti,In 4.1.3.1 is invariant underthe affine group

x, = x

y, =eQy

The general linear equation discussed in section 4.1.4 is invariant under thenonuniform-distortion group

XI = x

yi=y+a(x)

where (t) is any homogeneous solution of the equation.In each of these cases the group-theory method is entirely equivalent to

the method already introduced.

4.1.6 Exact Equations of the Second Order

We introduced the exact equation of the first order in section 2.2, Weshall now extend the ideas presented therein to the second-order differentialequation

F(y",y',y,x) = 0 (4-48)

We say that the differential equation (4-48) is exact if there exists a function4)(y , y, x) such that

F=dx

(4-49)

93


Suppose that such a functiGn GA exists. Then every solution of equation(4-48) must also be a solution of the first-order differential equation

4 (y', y, :t) (4-50)

for some constant CI. Hence, the problem of solving equation (4-48) can bereplaced by the problem of solving the first order equation (4-50). We cantherefore say that equation (4-48) has been reduced to the first-order equation(4-50). Upon recalling the definition given in section 1.3 we see that 4 is a(first) integral of equation (4-48).

It follows from the chain rule that

dx ax ay ay' (4-51)

where the variables x, y, and y' are treated as independent in forming thepartial derivatives. Hence, it follows from equation (4-49) and the fact thatdoes not depend on 5 " that the differential equation (4-48) must be anequation of the form

F ()/' , y' , y, x)=f (x, y, y' )3, g(x, y,y' ) = 0where 41

(4-52)

f(x,y,p)=Op (4 53)

g(x, y, p)= cfaz+ pch (4-54)

It is now easy to verify by substituting in equations (4-53) and (4-54) that fand g satisfy the conditions

f." + 2Pfry -4- P2fyy= grP± Pfi.11Pgi,

frp+ pfup + 2.6= gpp

As usual, we have put pm Y . The subscript notation for partial derivatives is being used.

94

(4-55)


We have therefore shown that, if the general second-order equation(4-48) is exact, it must be of the first degree and, hence, of the form (4-52).And the functions f and g in this equation must satisfy conditions (4-55).Then the equation will have a first integral 4. (and therefore can be reducedto a first-order equation) which can be found by integrating equation (4-53)with respect to p (at constant x and y) to obtain

43= f f (x, y, p)dp- h(x, y) (4-56)

where h is an arbitrary function which arises from the integration. Now ifthe differential equation is exact, it will always be possible to determine thefunction h so that, when equation (4-56) is substituted into equation (4-54),the latter equation will be identically satisfied.

In order to illustrate the method, consider the equation

F(Y1, y', x) xyY1+ xy12 yy' =0

For this equation the functions f and g (in eq. (4-52)) are given by

f= xy

g= ty'2+ yy' = xp2 F yp

(4-57)

(4-58)

It is easy to verify that these relations satisfy canditions (4-55). Hence, equa-tion (4-57) is an exact equatioa.

Substituting the first of equations (4-58) into equation (4-56) shows that

pxy+ h(x, y)

and substituting this and the second equation (4-58) into equation (4-54)shows that

xp2+ yp=yp-F h.,- F p2x- F phi,

But this equation is satisfied by taking h= constant. Hence, equation (4-57)

95


has the first integral41. pxy= y' xy= C1 (4-59)

where C1 is a constant. But this first-order equation is separable and can beintegrated immediately to obtain the general solution

2=Cilnx+C22

of equation (4-57).Sometimes a first integral of an exact equation can be found by inspection

simply by collecting terms and writing the equation in the form (4-49). Thus,since

,2YY =3, +YY

the differential equation (4-57) can also be written as

,F(Y% x) -x d

( YY ) (31) =0

Or

Hence,

F(y', y, x) xyy' = 0

= xyy'

which is the same as equation (4-59).The general linear equation (4-3) is of the form (4-52) with

(x, y, ') = Po(x)

and

g(x, y, y') = pi(x)y' + P2(9Y P3(.-0

96


The second condition (4-55) for exactness is automatically satisfied; and thefirst condition is satisfied if, and only if,

d2po dpidx2 dx

And when this is the case, equation (4-56) becomes

40= po(x)p + h(x, y)

But substituting this into equation (4-54) shows that

(((1a) _Lab\ ah

dx Pi .7- ay) P2Y (Tx- 1)3

Hence, it follows from equation (4-60) that

a [(dpo a R dpo ll(T): p y + h] p + (Tx- p y + hJ + p3 = 0

cir

It is easy to see that this equation will be satisfied when h is given by

h =dx

(dpop1) y f p3 ( x ) dx

(4-60)

Hence, we have shown that the first integral of the general, linear, exactequation is

= 1)0(x)yi (IP°dx y p3(x)dx =

This is a linear first-order equation and can, therefore, be solved by the methodsof section 2.3.

ins; as in the case of first-order equations, we can sometimes find anintegrating factor Ti(x, y, y') for an equation of the form (4-52) which is notexact. Thus, a function ri(y', y, x) is an integrating factor of equation (4-52) if

97


77F = 0

is an exact equation.For example, Liouville's equation

+g(y) y'2+f(x)y' = 0

is not exact; but, upon multipling through by 7 1/y1, we obtain the equation

which can be written as

1--; y"+g(y)y` +f(x)=-- 0

y' + Y(y)+X(x)] = 0

where Y( y) = f g(y)dy and X(x) = f f(x)dx.Equation (4-61) is therefore exact and its first integral is

41=ln y'+Y+X=Co

But, upon putting C1= eco, we obtain the separable equation

y'ev=Cle-x

which can be immediately integrated to obtain the general solution

felon dy= e-x(x)dx+ C2

(4-61)

to Liouville's equation.Let po(x), pi (x), and P2 (X) be the coefficients of the linear homogeneous

equation

98


PoY1 + ply' +p2y= 0

and let i (x) be a nontrivial solution of the equation

Per+ (2PO P1)71' + (14 Pi+ p2).71= 0

which can also be written as

(4-62)

(4-63)

*or NAY + np2= 0 (4-64)

Since equation (4-60) is a necessary and sufficient condition that equation(4-3) be exact, equation (4-64) shows that the equation

(71Po) ± (71P 1) ± (V32) Y

is exact. But this implies that v is an integrating factor for equation (4-62).The linear homogeneous equation (4-63) for the integrating factor 71 is knownas the adjoint equation of equation (4-62).

4.2 EQUATIONS WHICH ARE EQUIVALENT TO A LINEAR EQUATION

4.2.1 Equations Which Can Be Transformed Into a Third-Order Linear Equation

In section 2.4, we have seen that the first-order Riccati equation couldbe transformed into a second-order linear equation by the change of variable

y(oh(r)d.e (4-65)

where h(x) is not identically zero in the interval of interest.Since linear equations are generally much easier to solve than nonlinear

equations, the additional complexity incurred by raising the order of the equa-tion is often justified. We shall now show that the change of variable (4-65)transforms the second-order nonlinear equation

P(x)Y" 3h (x)YY =f(x) + g(x)y+ [h' (x) + p(x)h(x)iy2 h2(x)y3

(4-66)

99


where p(x), h(x), and g(x) can be any functions of x, into a linear equationof the third order. To this end we differentiate equation (4-65) three times insuccession to obtain

u' =- yh(x)u

u" = (y2h2 y'h + yh' )u

(4-67)

(4-68)

u"' =- h(y" 3hyy' + 2h' y +h" y-3y2le + h2y3)u (4-69)h h

Substituting for y" from equation (4-66) into equation (4-69) gives

u"' = p)y' + + g)y+f+ (ph 2h')du

And using equation (4-68) to eliminate y' in this equation gives

2h')u h" 2h'2 h')u"' +(p-- + hfu-- hyu(T+g- 17-+ p

Hence, it follows from equation (4-67) that u satisfies the third-order linearequation

u"' + pi(x)u" + qi(x)u' + ri(x)u=0 (4-70)

where

pi(x)=p-2h

h" h'2 h'q'xi= h h2 7-P -11-

(x) = hf

100



y"+3yy'+y3=0 (4-71)

This equation is of the form (4-66), with p=f=g=0 and h= 1. Hence,pi = qi = = and equation (4-70) becomes

u"' =0

But this equation has the solution

u Co + Cix + C2x2

which can be substituted into equation (4-67) to show that equation (4-71)has the general solution

2x +Y x24- boc-i-

where we have put bi =Cl/C2 and bo= Co/C2.

4.2.2 Equations Which Are Equivalent to a Second-Order Linear Equation

We shall now consider a class of second-order nonlinear equations whosesolutions can be expressed in terms of the solutions of second-order linearequations. The work on this problem began with Painleve (ref. 11) and wascarried on by Herbst (ref. 12), Gergen and Dressel (ref. 13), and Pinney (ref. 14).We shall present only the results here without proving any of the assertions.The references given should be consulted for details. Thus, for any functionsw(x) and q(x) of x, any function Y(y) of y, and any constant a, the differentialequation

1 Y' (y) w'(x)y"Y(y)y) y = [q(x) +aw2(x) exp 4 f

y(y)) Y(y)

(4-72)

has a solution of the form

y= O( &/2) (4-73)

101


'here 0 is the inverse of the function 0 defined by

d0(y) = exply(yy (4-74)

(that is, 0[0(y)] = y) and the function co is given by

= Cid; + C24 + C3u1u2 (4-75)

where C1, C2, and C3 arc constants and ul and u2 are two linearly independentsolutions of the linear differential equation

u w/ (x) ,u ' q(x)u =0 (4-76)

For example, consider the equation introduced by Painleve (ref. 11)

y"+b y'2+f(x)y' = g(x)y (4-77)

where b is a constant and f and g are any functions of x. This is a special caseof equation (4-72) with (1 Y')/Y= bly, w' /w = f, a =0, and q= (1+ b)g(x).Hence, we can take Y= y/(b+ 1) and equation (4-74) becomes

0= expkb + 1)dy =exp[(b+ 1) In = yo+0

Therefore,

0(0) = 011(b +1)

In this case the differential equation (4-76) is

u" +fu' (1+ b)gu= 0

And the solution to equation (4-77) is given by

102

(4-78)


0((41/2) = (w "2)11(1 +b) = (C ± C214+ C3111122) 1 /[2(1 +b)]

(4-79)

where a1 and u2 are two linearly independent solutions of equation (4-78).It can be verified by direct substitution and by applying equation (4-78) thatequation (4-77) will be satisfied if, and only if,

C3= 2\431

Hence, equation (4-79) becomes

y= (Vrlui + V u2)11(14-b)

But since VC; ui + VC; u2 is also a solution of equation (4-78), this is equiva-lent to taking y = where u is a general solution of equation (4 78).

Next consider the equation (ref. 14)

y" + p(x)y+ My-3=0 (4-80)

where p(x) is any function of x and M is a constant. This is a special case ofequation (4-72) with w= constant = Co, Y= y, q(x) = p (x) , a= mics.Hence, 0(y) =,y and, therefore, its inverse is 0(0) = 0. But this shows that

0 (0)1/2 = (01/2 = (COI ?-1- C24+ C3141U2) 1/2

where a1 and u2 are linearly independent solutions of

u" + p(x)u=0

Let J denote the Wronskian of the solutions u1 and u2; that is,

J= uit4 u;u2

(4-81)

(4-82)

(4-8.3)

Upon differentiating equation (4-83) with respect to x and using the fact thatui and u2 satisfy equation (4-82), we find that42

4t Or see section 5.9.

103


dfdx

which shows that f = constant. Substituting the solution (4-81) into equation(4-80), using equation (4-82), and using the fact that its Wronskian is constant,we find that equation (4-81) is a solution of equation (4-80) if, and only if,

C3 =2 (7-72-+cic2)

4.3 MISCELLANEOUS METHODS

4.3.1 Change of Variables

Frequently, a good choice of new dependent and independent variableswill convert an equation to a simpler and more easily analyzed form. Some pro-cedures which have proved helpful for this purpose are given in this section.

4.3.1.1 General transformation of linear equations. The second-orderlinear equation (4-3) is transformed by introducing a new dependent variablev which is of the form

v = f(x)y (4-84)

and a new independent variable t which is of the form

t=g(x) (4-85)

into an equation which is also linear and of the second order.The change of variable (4-84) transforms the linear homogeneous equation

y" + p(x)y` + q(x)y= 0

into an equation

(4-36)

v"+p, (x)vi +qi(x)v=0 (4-86)

of the same type. It is clear that equation (4-86) can transformed back into

104


the original equation (4-36) by a change of variable which is also of the form(4-84). Any two equations which can be transformed into one another by achange of variable of the type (4-84) are said to be equivalent. The new coeffi-cients pi (x) and q1 (x) of equation (4-86) are related to the original coefficientsp(x) and q(x) by

1 , 1 1 , 1 .

71P2=q1Y171:P1

Hence, we may say that the quantity

1 , 1(4-87)

remains invariant under the transformation (4-84). It is therefore called theinvariant of equation (4-36). We have seen that any two equivalent equationshave the same invariant. It can also be shown that, conversely, any two equa-tions with the same invariant are equivalent.

If the solution of one equation of the form (4-36) is known, it is possibleto find the solution of all equations which have the same invariant. Thus, whena new equation is encountered, we can compare its invariant with those ofequations whose solutions are known. If we can find one which is the same,we will have succeeded in solving the original equation. In particular, the ad-joint equation (see section 4.1.6) of equation (4-36) has the same invariantas equation (4-36) and is therefore equivalent to it. _

If the coefficient p(x) in equation (4-36) is identically equal to zero, we saythat the equation is in normal" form. In this case the invariant (4-87) is given by

5=q

In the general case the equation (4-36) with invariant (4-87) can always betransformed into the normal equation

v"+5v=0 (4-88)

by a change of variable of the form (4-84). Then equation (4-88) is said to be the

'Notice that in this case the meaning of term normal is different from that of section 1.4. The proper interpre-tation of this term should always be clear from the context.

105


normal form of equation (4-36). It is clear that any given equation has only onenormal form and that all equivalent equations have the same normal form.

The normal differential equation

d2yx2+.51(x)y= 0

can always be transformed into the normal equation

c121)

t2+J(t)v=0

by changing both the dependent and independent variables by transformationsof the types (4-84) and (4-85), respectively, provided the nonlinear differentialequation

(ci;dtp(t).(cw) d2tdt 2.5/Ux\ _1 d3t dt +3_ (I 2 dx3 dx 4 dx2

can be solved for the new independent variable t. When this is the case, thefunction f in equation (4-84) which determines the new dependent variable is

f(x) =dx

A fuller discussion of this topic as well as proofs of the various assertionsmade in this section can be found in Rainville (ref. 15, chapter 1).

4.3.L2 Transformation to an equation with constant coefficients. Wehave seen that the homogeneous Euler equation (section 4.1.3.2) can be trans-formed by a change of independent variable into an equation with constantcoefficients (which we know how to solve). More generally, the linear homo-geneous equation

y" +p(x)y' + q (x)y= 0 (4-36)

can be transformed into a linear equation with constant coefficients by a changeof independent variable if, and only if,

106


dq+ 2pqdx

= constant(1312

When condition (4-89) holds, the change in variable (4-85) is given by

4-89)

t=c f Vdx (4-90)

where c is any constant.In order to prove this, it is only necessary to substitute the change in vari-

able (4-85) into equation (4-36) and obtain the transformed equation

d2t. dtd2Y +dx21-13 dx dy gdo. dt dt

olx)

y -0

It is now easy to see that the coefficient of y in this equation will be constantif, and only if, t is given by equation (4-90). Further, substituting (4-90) intothe coefficient of dy /dt, we find that this coefficient will also be constant if,and only if, condition (4-89) is satisfied.

Thus, for example, for the homogeneous Euler equation, p(x)=polxand g(x) = qo /x2. Hence, equation (4-89) is satisfied, and equation (4-90) forthe new independent variable becomes in this case

no/2C dx= cqoo In x

x

The change of variable (4-30) given in section 4.1.3.2 is a special case of this.4.3.1.3 Interchanging of dependent and independent variables. -Differ-

entiating the identity

tiz

dx WY)

with respect to x shows that

488-942 0 - 73 8

107


d2y d2x dx\ -3dx2 dye 5(

And when these relations are substituted into the general differential equation(4-1), we obtain the equation

F [- (12 cd xdy2 \dy

A)- ( x1= 0

in which x is the dependent variable and y is the independent variable. Some-times this change of variable will result in an equation whose solution isknown or can be found.

4.3.1.4 Legendre transform. The Legendre transform which consistsof introducing the new independent and dependent variables p and q, re-spectively, defined by

dy

dyq = xdx- y

(4-91)

(4-92)

can be used to radically alter the form of a differential equation. It followsfrom these relations, after differentiating equation (4-92), that

dq= xdp+ pdx- dy dy= pdx

And when these equations are solved for x and y, we find that the inversetransformation is given by

dx =

q

dp

d q

-43 q

But it follows from equations (4-91) and (4-93) that

108

(4-93)

(4-94)


d2y dx d2qdx2 dp dp2

(4-95)

Hence, substituting equation (4-91) and equations (4-92) to (4 -95) intoequation (4-1) yields the transformed differential equation

F ((ch)-1 = 0dp2 ' P' P dp dp

which is also a second-order equation. If a solution q = f(p) of this equation canbe found, it can be substituted into equations (4-93) and (4-94) to obtain theparametric equations

Y= Pf' (P)-f(P)

S (P.)

of a solution y of equation (4-1).

4.3.2 Equation Splitting

When equations which are split in some natural way into sums, quotients,or products of terms, such as

or

f(y", y', y, x)+ g(y" , y', y, x) = 0

f(Yi, rI, y, x)g()P, y', y, x)

are encountered, it is sometimes possible to obtain a solution by putting

f(?, y', y, x) = h(x) = g()/' , y' , y, x)

for equations of the first type and

109


f()P, yi, y, x) = ch (x),

g( y' , y, x) = h(x)

for equations of the second type. If h(x) can be chosen so that a commonsolution to the pair of equations can be found, this solution will also be asolution of the original equation.

4.3.3 Tables of Differential Equations and Solutions

Two valuable catalogs of solutions to differential equations can be found inthe volumes by Murphy (ref. 16) and Kamke (ref. 17). Murphy lists over 2000solved equations which are classified according to order and degree; Kamkegives about 1500 equations along with their solutions and references to theliterature.

CHAPTER 5

Review of Complex Variables

A general procedure for solving second-order linear equations will begiven in chapter 6. But this procedure involves the use of power series, andthe discussion of power series becomes much simpler when it is carried outin its natural settingthe complex plane. In order to take advantage of thisfact we shall extend the definition of a differential equation given in chapter 1to include the case where the variables which occur in the equation are complex.Thus, by considering a more general situation we are actually able to simplifythe treatment. Another reason for making this extension is that it allows us tosee how solutions are connected across the singular points of the equation.

In this chapter, those concepts from the theory of functions of a complexvariable which ire needed for this purpose will be reviewed. The treatmentis essentially descriptive; and rigorous proofs of the various assertions are,for the most part, omitted. For a more detailed treatment of the topics coveredherein (including the omitted proofs), as well as a more complete coverage ofthe vast field of complex variables, the reader is referred to the many excellent.texts 44 which are devoted entirely to this subject.

5.1 COMPLEX VARIABLES

Let x and y be two independent real variables and, as is the usual practice,put i = V-=-1. Then z= x+ iy is a. complex variable. It is frequently convenientto think of the values of z as points in a plane, called the complex plane, whoseCartesian coordinates are x and y.

"A good elementary treatment is given (by Churchill (ref. 18). A more advanced and theoretical treatment isgiven by Ahlfors (ref. 19), while the text by (Carrier, Krook, and Pearson (ref. 20) emphasizes advanced applications.

111


FIGURE 5-1.Polar representation of complex number z.

Instead of using the Cartesian coordinates x and y, we can also use thepolar coordinates r= Nix2+y2 and 0= tan-1 y/x to locate points in this plane.(The relation between the polar and Cartesian coordinates is illustrated infig. 5-1.) Then

z= x+ iy= r cos 0+ ir sin 0

And upon using Euler's formula, we obtain the polar representation

z= r (cos 0 +i sin 0) = ref°

of the complex number z. Notice that for n= 0, ± 1, ± 2, . . .

ei2n.=

Hence,

cos 2rtr+ i sin 2n7r= 1

re= rei9e i2n7r== rene+2n7r)

The definitions of a domain and a neighborhood have been given insection 1.1 for a general n-dimensional space. We shall continue to use thesedefinitions in the two-dimensional complex plane,

112

REVIEW OF COMPLEX VARIABLES

The complex conjugate of the variable z is denoted by z* and defined byz*=x iy.

The absolute value or modulus of the complex variable z is defined to bethe length of the vector joining the origin with the point z in the complexplane and is denoted by Izi. Therefore,45

Iz I x2 ± y2 = VZZT

5.2 ANALYTIC FUNCTIONS OF COMPLEX VARIABLE

Let u(x, y) and v(x, y) be any two real-valued functions46 (of the variablesx and y) which are defined in some region of the complex plane. Then w = u+ ivis a complex - valued function of x and y. Since w associates a complex numberwith each point z=x+iy of some region of the complex plane, we say thatw is a function of the complex variable z.

We shall consider only a particular class of functions of a complex variablecalled analytic or holomorphic functions. In order to define this class of func-tions we first introduce the concept of a complex derivative. To this end, letw = u + iv be a function of the complex variable z and suppose that x and y arechanged by the amounts Ax and Ay, respectively. Then w changes by an amount

+ i Av. Now, by analogy with the definitions of the derivative of areal-valued function of a real variable, we define dw /dz, the derivative." of wwith respect to z at the point z, to be the limit

dw = limdu + i Av=

limAw

dz Ax+ iAy Az

provided that this limit not only exists but that it is independent of the mannerin which Ox and Ay approach zero. When dx and Ay approach zero in someprescribed manner, Ltz. approaches zero along some path in the complex plane,

*Notice that Izi is equal to the polar coordinate r.1" Recall that according to the convention adopted in section 1.1 we assume that all functions are single valued

unless explicitly stated otherwise.We shall frequently write u'(z) or tc' in place of dudz and u")(z) or u") in place of d^t /dan for n = 1,2,. . .

1 1 3


y

z + Az

X

FIGURE 5-2. Typical path along which Az can approach zero.

as indicated schematically in figure 5-2. But the definition implies that, ifw is to have a derivative at the point z, then Aw/Liz must approach the samelimit for every such path along which Az approaches zero. Although thisrequirement imposed on the complex derivative may seem unimportant, itsimplications are enormous. In fact, by allowing Az to approach zero alongvarious paths, it can be readily shown (ref. 18, p. 34) that, if u and v are con-tinuously differentiable;" a necessary and sufficient condition for the ex-istence of the derivative dw /dz at the point zo is that u and v satisfy the twoCauchy-Riemann equations

au.avax ay

au avand =ay ax

at this point. This shows that much greater restrictions are imposed on thosecomplex functions which possess complex derivatives in the sense of thedefinition given above than are imposed on the real functions which possessordinary real derivatives. However, since the complex derivative is formallythe same as the real derivative, the usual rules for differentiating sums, prod-ucts, quotients, etc., still apply (ref. 18, p. 31).

A function w(z) of the complex variable z is said to be analytic4 9 or holomor-phic in a domain D if it possesses a derivative at every point of D. Frequently,

"This means that the partial derivatives aulax,aulay,av/ax, and avtay exist and are continuous." 1 he terms regular and monopole are also used.

114


when it is of no consequence in the discussion, the reference to the domain D isomitted and we simply say that the function w(z) is analytic. A function issaid to be analytic at a point zo if it is analytic in some neighborhood of thispoint.

Notice that the identity function

1.1)-= Z = iy

is analytic at every point since the Cauchy-Riemann equations (with u = x andv = y) are always satisfied. However, the complex conjugate of this function

w z* = X iy

is not analytic at any point since in this case au/ax = 1 and av/ay = 1 andtherefore the Cauchy-Riemann equations are never satisfied.

The real and imaginary parts of the function

are

1 1 z* x iyz x + ty zz* x2+ y2

U = x2 + y2 and v x2 + y2

respectively. And upon taking the partial derivatives of these functions we seethat the Cauchy-Riemann equations are satisfied at every point except z= 0,where the partial derivatives fail to exist. Hence, w = 1/z is analytic at everypoint except z = 0.

It is a remarkable fact (see ref. 18, p. 122) that any function w(z) which isanalytic at a point zo possesses derivatives of all orders at this point. And thesederivatives are themselves analytic functions at this point. It is easy to see thatthe sum and product of any two analytic functions are analytic within anydomain hi which both functions are analytic. Usually, any function which isobtained from a real algebraic, elementary-transcendental function (trigono-metric, exponential, logarithmic, etc.) or a common higher-transcendental func-tion (Besse' function, hypergeometric function, etc.) by replacing the realvariable x by the complex variable z=x+iy is an analytic function within somedomain.

115


5.3 CONFORMAL MAPPING 5°

According to the definition given in section 5.2, a complex functionw= u+ iv of the complex variable z=x+iy associates a pair of numbers(u, v) with each point (x, y) of some region of the complex z-plane. We caninterpret these numbers as coordinates of a point in a complex w-plane. Thus,we may think of the function w(z) as a transformation or a mapping of someregion in the z-plane into some region in the w-plane. More specifically, wecan think of an analytic function w(z) which is defined on a domain D in thez-plane as a mapping of D onto a region R in the w-plane, as shown schematicallyin figure 5-3. In order to determine the properties of an analytic function w(z)it is frequently helpful to study the manner in which this function transformsvarious points, curves, or domains in the z-plane into corresponding points,curves, or regions in the w-plane.

y

FIGURE 5-3. Mapping of D onto R by w(z).

Thus, the transformation w= 1/z associates a single point in the w-planewith each point in the z-plane except the origin z= O. It maps all those pointslying outside the circle with radius R and center at the origin in the z-planeinto the interior of the circle with radius 1/R and center at the origin in thew-plane. Although there is no point of the z-plane which maps into the pointw= 0, we can, by choosing R sufficiently large, make the points outside this

s" A comprehensive treatment of this subject can be found in Nehari (ref. 21).

I I 6


circle map into a circle of arbitrary small radius about w= O. However, it isfrequently convenient to "complete" the z-plane by adding a fictitious pointz =00 which maps into the point w =0 under the mapping w = 1 /z. Thus, whenwe speak of the behavior of an equation or a function at the point z=-- 00, weactually mean the behavior at the point w= 0 of the transformed equation orfunction obtained by putting z= 1/w. When it is necessary to distinguishbetween the point z= co and the other points of the complex plane, the latterare said to be finite points. When the complex plane includes the point z= 00,it is called the extended plane; and when z =00 is excluded, it is called thefinite plane.

Let w(z) be analytic in a domain D and let zo be any point of D at whichdw /dz 0 0. Then w(z) transforms any two smooth curves passing through zoin such a way that their image curves intersect at the point wo= w(zo) withthe same angle (in both magnitude and sense of rotation) as the original curvesin the z-plane (ref. 18, p. 174). Thus, the mapping "preserves angles," and wesay that w(z) is a conformal mapping at all points where dwl dz 0.

A mapping which is particularly useful for the treatment of certain typesof linear differential equations is the linear fractional transform

az + bcz+ d

(5-1)

Notice that, if ad bc= 0, this transformation reduces to w= constant. Inany other case, it transforms each point in the extended .z -plane into a pointin the extended w-plane in such a way that no two points in the z-plane map intothe same point in the w-plane. In addition, there is a point in the z-plane whichmaps into each point in the w-plane. For this reason, we say that a linear frac-tional transformation with ad be 0 0 is nonsingular.

If we consider straight lines and points as being degenerate circles (i.e.,circles having infinite or zero radii), we can say that the linear fractionaltransformation always maps circles into circles.

Performing two nonsingular linear fractional transformations in suc-cession is equivalent to performing a single nonsingular linear fractionaltransformation. In fact, any nonsingular linear fractional transformation canbe performed by carrying out not more than four successive transformationseach of which has one of the three elementary forms

w=- az (rotation and stretching) (5-2)

117


w= z (translation) (5-3)

1 (inversion). (5-4)

In order to prove this, first suppose that c 0 0. Then equation (5-1) canbe written as

( b\ ( ad\2w=.a

c

Now transform the z-plane into the ti-plane by the transformation

d\c

Then transform the ti-plane into the t2-plane by

1t2=-ti

and the t2-plane into the 43-plane by

b adt3=c C2

t2

Finally, transform the t3-plane into the w-plane by

aw= ,

t3

(5-5)

Upon combining these successive transformations we obtain equation (5-5)and therefore equation (5-1). This proves the assertion for the case wherec 0. When c=-- 0, equation (5-1) reduces to

118


a bw dz+ d

which is easily seen to be equivalent to the succession of transforms

ati=--c-iz and w=t1--Fi

If the transformation (5-1) is to be nonsingular, at least two of the con-stants a, b, c, and d must be nonzero. By dividing through by one of these,it is easy to see that equation (5-1) in fact contains only three arbitrary con-stants. It is therefore not surprising that these constants can always be chosenso that the linear fractional transformation maps any three given points in thez-plane, say Zi, z2, and z3, into any three given points, say ut, and w3,inthe w-plane. Thus, for example, the linear fractional transformation whichtakes z = zi into w= 0, z=z2 into w= 1, and z= z3 into w= is

Z2 Z3 Z Z1w=Z2 Z1 Z Z3

5.4 ISOLATED SINGULAR POINTS OF ANALYTIC FUNCTIONS

If a function w(z) is analytic at every point in some neighborhood of apoint zo except at the point zo itself,5' then zo is called an isolated singular pointor an isolated singularity of the function w(z). Thus, the function

z + 2w(z) z(z + 1)2 (5-6)

has isolated singulailties at the points z = 0 and z = 1.An isolated singular point zo of the function w(z) is called a removable

singularity if the limit of w(z) as z ) zo is equal to some finite number. Let k bea positive integer. An isolated singular point zo of the function w (z) is said to bea pole of order k if the limit as z zo of the quantity (z -zo) kw(z) is equal tosome finite nonzero number. A pole of order one is called a simple pole.

51The function need not even be defined at the point zo.

119


Thus, for example, the function given by equation (5-6) has a simple poleat z = 0 and has a pole of order two at z = 1.

Because of the way analytic functions arise in practice, it turns out thatwesometimes arrive at a function w(z) which is not defined at a point zo but isdefined and analytic at every point of a neighborhood of zo. The point zo isthus an isolated singular point of w(z). But if this point is also a removablesingularity, the function can be made analytic at zo simply by assigning a suit-able value to w(z) at this point (ref. 18, p. 158). For example, since division byzero is undefined, the function w(z) given in equation (5-6) is undefined atz = 0. Thus, 0 X w(0) is also undefined. Hence, the function 4(z) defined by

4(z) = zw(z)

is not defined at the point z = 0. However, upon defining (0) by

4(0) lizm 4(z) = 2

we obtain a function which is analytic at z = 0.It is easy to see that, if w (z) has a pole of order k at zo, the function

4(z) = (zzo)kw(z)

has a removable singularity at zo and is therefore "essentially" analytic at thispoint.52 We shall sometimes say that an analytic function has a pole of orderzero at the point zo if zo is a removable singularity of this function.

Any isolated singular point of an analytic function which is not a pole ora removable singularity is called an essential singularity. For example, thefunction sin(1/z) has an essential singularity at z= 0.

There are some important differences between poles and essential singu-larities. For example, if w(z) has a pole at zo, the function 1/w (z) is analytic 53

52 For the purposes of this book we can assume that any analytic function which is encountered has alreadybeen defined at its removable singularities in such a way that it is analytic at these points.

53 In fact, it is equal to zero at zo (see preceding footnote).

120


at zo; but if w(z) has an essential singularity at zo, so does liw(z) (ref. 22, p.110). A pole, then, is a point where a function w(z) is not analytic only becauseits modulus Iw(z) I becomes infinitely large at this point and for no other reason.

A function which is analytic at every finite point of the complex plane iscalled an entire function. And Liouville's theorem (ref. 18, p. 1.25) states thatany entire function which is also analytic at infinity (see section 5.3) must, infact, be equal to a constant.

A polynomial is a function of the form ao+ aiz + . . . + anzn whereao, . . . an are complex constants. It is an entire function, and it has a poleof order n at infinity. A rational function is the ratio of two polynomials (whichmay be chosen to have no linear factors in common). It is therefore a functionof the form

ao+ajz+ azz2+ . . . +anznu;(z) an 0 0; I), 0 0 (5-7)bo+ biz+.b2z2+ . . . bmz'n

and is analytic everywhere in the finite plane except at those points where itsdenominator is equal to zero. These points are poles of w(z). If m n, thenw(z) is analytic at the point z= 00; otherwise it has a pole of order n m atthis point.

A function which is analytic at every point of a domain D except at thosepoints of D where it has poles is said to be meromorphic in D. For example,the rational function (5-7) is meromorphic in the entire finite plane. In fact,any function which is meromorphic in the entire finite plane and has a pole atinfinity is necessarily a rational function (ref. 20, p. 60). A function which ismeromorphic in the entire finite plane can have at most a finite number ofpoles in any domain D o§ finite size. However, it may have infinitely manypoles if D is infinitely large.54

In view of Euler's formula (see section 5.1) it is natural to define the func-tion ez by the formula

ez= ex+ IY= ex cos y +lex sin y

Then et is an entire function and has an essential singularity at z= 00. We cannow define the functions sin z and cos z by the formulas

54 For example, D could be the entire plane or the upper half plane.

121


sinz--=eize-iz

2ieiz+e-it

and cos z2

which extend in a natural way the definitions given to these functions whenthe variables are real. They are also entire functions with essential singularitiesat z= 00. On the other hand, the function 1/sin z is meromorphic in the entirefinite plane; and its poles are located at the points z= nv for n=0, 7.4_- 1, -1.- 2,etc. It also has an 'essential singularity at z= 00.

5.5 POWER SERIES

A power series about a point zo is an infinite series of the form

an(zzo) nn=-0

(5-8)

in which the coefficients an can be any complex numbers. This series certainlyconverges at the point zo, which may be the only point at which it actuallydoes converge. Or the opposite extreme could occur and the series mightconverge at every point of the finite plane. In all other cases the series willconverge at every point within a circle of radius R and center at zo, called thecircle of convergence of the series, and will diverge at every point which liesoutside this circle. Thus (ref. 19), there exists a number R lying in the range

00 called the radius of convergence of the series such that the series(5-8) converges at all points z which satisfy the inequality 55 < R anddiverges at all points z which satisfy the inequality 56 Iz zol > R. The questionof whether the series is convergent for the points which satisfy the equalitylzzol =R (i.e., points on the circle of convergence) is more subtle but un-important for our purposes.

A power series with a nonzero radius of convergence R converges to ananalytic function and can be differentiated term by term (i.e., the order ofsummation and differentiation can be interchanged) at every point within itscircle of convergence. Thus, there exists a function w(z) which is analytic atevery point within the circle lzzol < R such that

"The series is, in fact, absolutely convergent at these points. This means that the (aeries still converges wheneach of its terms is replaced by its absolute value. See ref. 19 for more details.

M This result was first established by Abel.

122

w(o=n=0

dul(z)dz n=1

an(zzo)"

1nan(z zo)"1


for Iz zol < R

It can also be shown (ref. 19) that, when the series (5-81 has a nonzeroradius of convergence R, there exists a positive constant M co such that forany number 0 < r < R the coefficients of the series satisfy the Cauchy estimates

lard <Mrn

for n=0, 1, 2, .

This result will be used in the discussion of the solutions of differential equa-tions in the next chapter.

Now suppose, that w(z) is analytic in the domain D and that zo is anypoint of D. Then the power series

3` 1

n!W(n)(Z0) (Z Z0) n

n=0kJ 'I

(where woqzo)=----- w(z)) converges to w(z) at every point z within the largestcircle centered at zo lying entirely within D (ref. 18, p. 129). This series is calleda Taylor series expansion of w(z) about zo. Its radius of convergence is at leastequal to the shortest distance between zo and the boundary of D. It may belarger than this but we have no guarantee that the series will converge to w(z)at points which lie outside of D. The series representation (5-9) is unique in

the sense that if bn(zzon is any power series which converges to w(z)n=0

within any circle about zo, then necessarily (ref. 20, p. 49)

b,,=! w(n)(zo) for n=0, 1, 2, . .n1

Next, suppose that w(z) is analytic at every point of a domain D exceptfor a certain number of isolated singular points and let zo be a point of D atwhich w(z) is analytic. Then the circle of convergence of the Taylor series

123

488-942 0 - 73 - 9


X Isolated singular points of wrzt

(a)

Circle of convergence

rMaximum size/ of circle of

convergenceBoundary,- -of D

fbl

(a) Circle of convergence does not intersect boundary of D.(b) Maximum circle of convergence intersects boundary of D.

FIGURE 5-4.Circle of convergence of a Taylor series.

of w(z) about zo passes through the nearest isolated singular point of w(z)if it does not into the boundary of D. In this latter case we can only assertthat the radius of -!onvergence does not exceed the distance between zo and thenearest of the isolated singularities of w(z) within D. These results are illus-trated in figure 5-4. Thus, in particular, if the function w(z) is analytic at everyfinite point of the complex plane except at a certain number of isolated singu-lar points, then the circle of convergence of its Taylor series expansion aboutany point zo where w(z) is analytic always passes through the nearest isolatedsingular point to zo.

For example, we have seen in section 5.2 that w(z) = 9 is an entirefunction. It is easy to see that the nth derivative of this function is

124


ww(z) = ez for n=0, 1, 2, . .

Hence, the Taylor series expansion of ez about z= 0 is

x 1

!nezz

nn=0(5-10)

And this series converges in the entire finite plane.We have also shown in section 5.2 that the function w(z) = 14 is analytic

at every point of the complex plane except the origin z = 0. The nth derivativeof this function is

w(n)(z) =(-z1 )n-fni n!

Hence, its Taylor series expansion about the point z = 1 is

= etc ( (Z 1) nZ n=0

It is easy to verify that the radius of convergence of this series is equal to 1and that the circle of convergence passes through the isolated singular pointz = 0. Upon replacing I z by z in this series we obtain the geometric series

1

1 -z =E Znn=0

(5-11)

which converges within the unit circle IzI = 1. This circle passes through theisolated singular point z= 1 of the function (1 z) -1.

We shall frequently find it necessary to add and multiply two powerseries. The sum of two power series about the same point can be obtainedby adding the two series term by term. The resulting series will convergewithin the smaller of the two circles of convergence of the original series(ref. 23, p. 123). Now let cn and do be any two absolutely convergent

n= n=

series. Any expression for the product (26 cn) (26 cln) must certainly includen= n=

125


all terms of the type cidi. But all terms of this type must belong to the array

codo+ codt code +. . . coda +

ci do + + (12 dn .

+ C2do + C2 d1 C2d2 + . C2dn +

cmdo+ cmdi cmd2 . cnin+. . .

And the series which is formed by grouping together the terms along the di-agonals of this array and summing the result over all diagonals is called theCauchy product of the two series. Thus, the general term in the Cauchyproduct is

an = Cnd0+ Cn-Idl Cn-2d2 + . . COdn = Cn-kdk = Ckdn-k0 =0

00

and the Cauchy product is the series E an. Cauchy's theorem (ref. 18, p. 147)n=0

states that the Cauchy product an is absolutely convergent and thatn=0

(n=0 (2 (2Cn) (±c dn) = (±e = Ckdn-kn=0

The Cauchy product is particularly convenient to use when multiplyingpower series. For, in this case, we get

126

n

an(// Z Zo)nl [E bn(Z zO)n] =nte0 n=0 n=0 k7--0

ak (Z Zo)kbn-k (Z ZO)"

= E (Z. Zo) n akbn-k)n-0 .,c=0


which converges within the smaller circle of convergence of the two original.series.

For example, let us use these ideas to find the power series expansion ofthe function 1 /[(a z) (bz) where a and b are arbitrary complex constants.To this end notice that the geometric expansion (5-11) implies that

1

1 za

= E a-net andn=0

1 E1 -? n=°

Hence, upon forming the Cauchy product we obtain

But since

we find that

1 1 1c n n 1 k(a

zz)(b z) ab te,o(a

1

n 1 Nkt nk (11.)11+1 (6n+1)k4d_ok a) k b ) 1 -1

a b

1 1

(a z)(b z) b a ,&0[a--(n+1) b-(n+11zn

Next, in order to find the Taylor series expansion of the function cos z/(1+ z2)about the point z =0, notice that by changing variables in the geometric expan-sion (5-11), we obtain the expansion

1

1 + z2 Ln=o( 1)nz2n

and that by using the expansion (5-9) and the definition of cos z given in section5.4, we obtain the expansion

127


cost=.2n( (2n)!

Then, upon taking the Cauchy product of these two series, we obtain

n 1N- Wien (2k)tdo X0z

COS Z2

'e have seen that a power series (with positive exponents) represents(converges to) an analytic function within a circle. Similarly, the series (ref.18, p. 134)

Cle

E an(zzon (5-12)x

containing both positive and negative exponents, converges to an analyticfunction in an annular region 57 lying between two concentric circles centered.at zo and of radii RI and R2 with R1 < R2 (i.e., at all points z for whichRI < lzzo < R2). Conversely, any function w(z) . which is analytic in anannular region R1 < zol < R2 can always be represented by a series ofthe form (5-12) at every point of this region. This expansion is called a Laurentseries. An important special case occurs when the function w(z) has an isolatedsingularity at the point zo and is analytic at every other point within the circleiz zol = R. In this case the series

to(z)= E a n(z zo)"

converges to w(z) at every point z of the puncturcd circular region0< I z zo I <R bounded by the circle lzzo I =R and the point zo. Then theisolated singularity zo is a pole of order k of the function w(z) if, and only if,a_k 74 0 and an 0 for all n < k. Hence, if zo is an essential singularity,infinitely many negative powers of z zo will occur in the series.

For example, we can use the geometric series (5-11) to obtain the expansion

57 Prinided it comet-vs at ell.

128

1 1 1

,-1 -0z


-1= E z.

of the function (z 1)-1 about the point z = m. This series converges at allpoints which lie outside the circle Izi = 1. And since the series

3 =E 3-z3 z n=o

converges within the circle 1z1 = 3, we conclude that the series

with

2r. 1 3anzn

(z 1)(3 z) 1 3 z

=lln 3n for n= 0, 1, 2, . . .

n=x

{1 for n = 1, 2, . .

converges in the annular reWon 1 < 1z1 < 3.

5.6 COMPLEX INTEGRATION

Let w J(z) be analytic in a domain D. Then the integral w(z)d;, of w(z)r

along a curve or path 1' which lies entirely within D is defined in terms of tworeal line integrals along I' by

w(z)dz = dx vdy) + i (a dy + vdx) (5-13)

It is easy to verify from this definition that the usual rules of integrationstill apply. Thus, in particular, if a and /3 are complex constants and wi(z)and w2(z) are analytic functions

1:29


Jr[awl (z) + 13w2(z)]dz= wi(z)dz+ f w2(z)dz

and if the direction of integration along r is reversed, the integral is multipliedby 1.

For example, the integral of the function w (z) =z along the line yo = con-stant, from the point (0, yo) to the point (xo, yo), is

Iv

zo 1zdz=f xdx- F.ro

i yodx=---2X 2

iyox0=2

(x0+ iy0)2-1 (iyo)22

In order to integrate w(z) =z along a circular path centered at z=-- 0, it is bestto use polar notation. Thus, let the radius of the circle be R. Then on this circle

Rew and dz= iRei° d9. Hence,

f zdz= i Jo2.

.R2e2l°d0= 0Jr=R 0

Mors; generally, it can be shown (ref. 18, p. 111) that, if C is any closed curvewithin D and if w(z) is analytic at every point in the interior of C, then

w ( z ) dz -= 0

Therefore, if r, and r2 are any two curves in D which begin at the point zi andend at the point z2 (fig. 5-5), then

Jtv(z)dz=--w(z)dz

r2

provided w(z) is analytic within the domain enclosed by these curves.Because of this fact the exact path along which the integration is carried

out is frequently unimportant; and when this is the case, we writeJ

22

w(z)dz in place of f w(z)dz.

The function w(z) = 1/z has a singular point at the origin; and its integralalong the circle of radius R centered at the origin is

130

(a)


(b)

(a) Line integrals along F1 (b) Line integrals equal alongand I'2 not necessarily F1 and F2.equal.

FIGURE 5-5. Paths for line integral. (Arrows indicate direction of integration.)

1 f 27rdz= d0= 27riL=R Z 0

If the function w(z) is an analytic function of z in D, the function F(z) de-fined by

F(z) w(z)dzz0

is also an analytic function of z in D and (ref. 18, P. 114)

Hence,dz

dF (z)= w(z)

F(z) F (zo) = F' (z)dzzo

This shows that just as in the case of real variables, integration and differen-tiation are inverse processes. For example, in order to evaluate the integral

fZ

zz"dz notice that, for n 1,

131


Hence,

1 dz"+izn= n +1 dz

1 iz dzn+1 zn+i zon+izndz= n+ 1 zo dz

dz n+ 1 n+ 1zo

It can also be shown (ref. 18, p. 141) that any convergent power series canbe integrated term by term and the resulting series will have the same circleof convergence as the original series. In fact, it can be shown that the productof any convergent power series about a point zo with a function of the form(z zo)A, where is a complex constant, can also be integrated term byterm.

5.7 ANALYTIC CONTINUATION

5.7.1 Definition

First, suppose that w, (z) and w2 (z) are both analytic in some commondomain D. It can be shown (ref. 18, p. 259) that if wi(z)= w2(z) at all points ofsome subdomain of D or even at all points of some curve which lies entirely withinD, then wi(z) and w2(z) are equal at every point of D. This assertion is knownas the fundamental theorem of analytic continuation. It means that there isonly one analytic function in a domain D which takes on any given set of valueswhich are prescribed at every point of a subdomain of D or even at every pointof some curve in D. For example, if the function w(z) is analytic in a domain Dand is equal to zero at every point of a subdomain of D or at every point of acurve lying within D, then w(z) is zero at every point of D.

We have seen in section 5.5 that the analytic fur:ction

1

1 z(5-14)

which is defined and analytic at every point of the complex plane exceptz =1 can be expanded in the Taylor series

cc

E z"n=0

132

(5-15)


1111111110

9111iliF

FIGURE 5- 6. Domains for direct analytic contination; D, is the common part of both domainsDI and D2.

which converges only within the unit circle. Thus, the function (5-15) is de-fined only within the unit circle, where it is equal to the function (5-14),which, however, is defined in a much larger region. The function (5-14) cantherefore be considered as an extension of the function (5-15) from the unitcircle to the entire complex plane with the point z = 1 excluded. Indeed,whenever an analytic function is defined by some expression (such as a powerseries) in some domain D which is not the whole complex plane, it is naturalto ask if this function can be extended to a larger uomain.

First, consider the function w1(z) defined on the domain DI and let D2be another domain, part of which coincides with D1 as shown in figure 5-6.It can be shown that the common region Dc, which is part of both domains,is itself a domain 58 and hi ;ice is a subdomain of both the domain D1 and thedomain D2. Now suppose that there exists a function w2 which is analyticin the domain D2 and which is equal to w1 (z) at every point z of the commondomain Dc. Of course, the function w2 (z) may not exist. However, if it doesexist, it is called the direct analytic continuation of the function w1 to thedomain D2.

There can be at most one direct analytic continuation of a function to anygiven domain. For if 4 is another direct analytic continuation of wi to D2, thenw2 and 42 are both analytic in D2 and are equal to one another in the subdomainDc. Hence, the fundamental theorem shows that w2 and 42 are equal at everypoint of D2. And this of course means that 4 and w2 are the same function.

"See, e.g., ref. 8, ch. 1.The domains DI and D2 are said to intersect; and the common domain Dc, called theintersection of DI and D2, is denoted by DI fl D2.

133


FIGURE 5 -7. Chain of domains for analytic continuation.

Let b be the domain which consists of all points which belong either tothe domain D1 or the domain D2, or both.59 Then since the analytic functionswi (z) and w2 (z) are equal at all points where they are both defined, we candefine a new analytic function iv(z) on the domain D by the relation

(z)iv(z) --=

w2(z)

for z in D1

for z in D2

It is clear that the analytic function iv (z) is an extension of the analytic func-tion w1(z) from the domain D1 to the larger domain D.

The process described above does not have to terminate with the function(z). It may, for example, be possible to find a direct analytic continuation

w3 (z) of the function w2(z) to a domain D3, and so on. Proceeding in thismanner we obtain a chain of domains D2, D3, . . . such as that shown in figure5-7 and a collection of analytic functions w2 (z) , w3 (z) , . . defined on thesedomains. Each of these functions is said to be an analytic continuation ofthe function wi (z), and the procedure itself is called analytic continuation.We say that wi (z) is analytically continued along a simple "'curve F which ex-tends from D1 to some point P if F is completely covered by a chain of domainsD2, D3, . . . (as shown in fig. 5-8) along which w1(z) can be analytically con-tinued in the manner described above.

" It is shown in various books on analysis that this extended region is indeed a domain. It is called the union ofthe domains DI and D, and is denoted by DI U D2, e.g., see ref. 8, ch. 1.

6° Roughly, this means that r is smooth and does not cross itself nor have any other pathological behavior.

134


FIGURE 5-8. Analytic continuation along a curve.

The collection of functions generated by the process of analytic continua-tion can be used, as in the case of direct analytic continuations, to define anew analytic function on the larger domain which includes all the domainson which the analytic continuations are defined. If, in addition, the functionw1(z) is analytically continued along a curve F, the values of this extendedfunction on F itself will be the same no matter which specific collection ofdomains D2, D3, . . . is used to construct it.

5.7.2 Specific Method

In order to make these ideas more concrete we shall consider a specificprocess which can be used, at least in theory, to obtain an analytic continua-tion of any given function. Thus, suppose that the analytic function wl (z) isdefined by some expression in the domain D1. For example, it may be definedby a Taylor series

wl (z) = an (z z1)71 (5-16)n=0

in which case the domain DI will be the interior of a circle of radius R 1 centeredat the point zi. We suppose that R 1 is finite. Choose a point z2 in the domain DI.Since an analytic function is infinitely differentiable, we can calculate thesequence of derivatives wcn)(z2) for n =0, 1, 2, . . . from the given expressionfor wi (z) in the domain DI. For example, when the function wl (z) is given bythe Taylor series (5-16), /40(z2) can be obtained by differentiating equation(5-15), term by term, n times and evaluating the result at z2. Then as indicatedin section 5.5 the Taylor series

135


with

w2(z)= E b,,(z z2)"n=o

u4n)(z2)b for n=0, 1, 2, . .n!

will converge to an analytic function in a circle of nonzero radius centered atz2. And this function will be equal to w1(z) at every point which is inside boththis circle and the domain DI. Of course, the circle of convergence of w2(z)may not extend beyond the domain DI (as shown in fig. 5-9(a)). If this occurs,

136

Circle of convergenceof w2(z)

(b)

rCircle of convergenceof w2(z)

(a) Circle of convergence of w2(z) does not extend beyond DI.(b) Circle of convergence of w2(z) extends beyond DI.

FIGURE 5-9.Analytic continuation by power series.


we can choose a new point z2 and repeat the process. However, if the circleof convergence of w24) does extend beyond D1 (as shown in fig. 5-9(b)),then w2(z) will be an analytic continuation of wi (z). The process can now berepeated by choosing a point za which lies within the circle of convergence ofw2 (z)., proceeding to obtain a new Taylor series about this point, and so on.

It can be shown that any analytic continuation of a given function, nomatter how it has been obtained, can also be found by using the method ofpower series just described. It is easy to verify from this that the analyticcontinuation of the derivative of an analytic function has the same value atany given point as the derivative of the analytic continuation at that point,provided they are both carried out along the same curve. This means that theorder of differentiation and analytic continuation can be interchanged.

5,.7.3 Singular Points

Let w(c) be analytic at all points of a domain D except for a certainnumber of isolated singular points. Suppose, in addition, that w(z) is analyticon a subdomain D1 of D and that w1(z) is the restriction61 of w(z) to D1. Thenwi(z) is analytic on D1 and can be analytically continued to any other sub-domain of D which does not contain singular points of w(z). As long as theanalytic continuation of wi(z) is carried out along a path which lies entirelywithin 1), the value of this analytic continuation at any point of D will be equalto the value of w(z) at that point. However, since the circle of convergence ofa power-series expansion of an analytic function will pass through its nearestisolated singular point (provided that point is nearer than the boundary of thedomain), it can be seen by using the method of power series that the functionwi (z) cannot be analytically continued along any curve which passes throughan isolated singular point of w(z). (These ideas are illustrated in fig. 5-10.)

More generally, let ws (z) be analytic on some domain D1. If this functioncannot be analytically continued along any simple curve which crosses theboundary of D1 at the point zo, we say that the point zo is a singular point ofthe function wi(z). And the preceding remarks show that this definition isconsistent with the definition of an isolated singular point given in section 5.4.

Let E an (z zo). be a power series whose radius of convergence is notn=0

61 That is, wo(z) is a function which is defined only on and takes on the same values at each point of DI as the

function u(z)

137


Analytic continuationof w1(z) along thispath is equal to wtzl-\

X Isolated singular points

r Analytic continuationcannot be carried out

/IIpast this point

FIGURE 5-10. Illustration of analytic continuation of a restriction of an analytic function.

w tzl

rEnlarged circle C/ obtained by analytic

Untinuation of Mil

/Circle of convergenceof original series

FIGURE 5-11.Analytic continuation past a circle of convergence.

138

..


equal to zero or infinity. We have seen that this series converges to an analyticfunction w(z) everywhere Within its circle of convergence and diverges outsidethis circle. We shall now show that this circle always passes through a singularpoint 62 of w(z). In order to obtain a contradiction, suppose that there were nosingularities of w(z) on the circle of convergence. Then it would be possible toanalytically continue w(z) a finite distance outside this circle everywherearound its circumference (as shown in fig. 5-11). These analytic continuationscould then be used to construct an analytic function wi(z) which is an ex-tension of w(z) to a larger circle C which is also centered at xo. Now it is shownin section 5.5 that the circle of convergence of the Taylor series expansionof wt (z) about zo cannot be smaller than C. But it is also shown in that sectionthat this extended function iv' (z) must have the same Taylor series expansionabout zo as w(z). However, this is impossible since (by hypothesis) this latterseries diverges outside of the smaller circle. Hence, we must conclude thatthere is a singular point of w(z) on its circle of convergence.

Starting with a given analytic function wi (z) defined on a domain DI, wecan carry out the process of analytic continuation until all possible analyticcontinuations of the function wt (z) have been found. The collection of analyticfunctions generated in this manner can again be used to define a new functionon the domain which consists of all the domains on which these various func-tions are defined. The function obtained in this manner is called a completeanalytic function. This function cannot be further extended. A point zo is saidto be a singular point of the complete analytic function if it is a singular pointof any analytic continuation of w1 (z). And sometimes, when no confusion islikely to arise, we shall say that zo is a singular point of the original functionwt (z) itself.

5.7.4 Multiple-Valued Functions

There is a certain difficulty associated with the definition of a completeanalytic function given in the preceding section. Thus, suppose that theanalytic function wi (z) defined on the domain DI can be analytically continuedalong the two simple curves ti and F2 which terminate at the same point p,

'In section 5.5 we only asserted that the radius of convergence does not exceed the distance between zo and thenearest isolated singular point.

139

488-942 0 - 73 - 10


Singular point

FIGURE 5-12. Paths for analytic continuation of multiplevalued function.

'-No branch points ofwilzi in this region

ANNE/ANIIINEV

AN EEELANIINANNEbh,

. INNIPITINNIESAMINANWINNINIIN

FIGURE 5 -13. Domain for analytic continuation.

140


as shown in figure 5-12. There is no guarantee that the analytic continuationof wt (z) along r will have the same value at p as its analytic continuationalong I'2. But if this occurs, the complete analytic function obtained fromwt (z) must have more than one value at the point p. Thus, although up to thispoint we have assumed that all functions are single valued, we must in generalallow a complete analytic function to be multiple valued. It can be shown'ref. 19, p. 218) that the analytic continuation along the path fi will alwayshave the same value at the point p as the analytic continuation along F2 unlessthere is a singular point (such as that shown in fig. 5-12) between these twocurves.° However, the mere existence of a singular point between the twocurves does not guarantee that the analytic continuations along the two dif-ferent curves will have different values at p. This only occurs when the singularpoint is a branch point. A branch point is a singular point of a function whichhas the property that the function will not return to its starting value uponanalytic continuation along any arbitrarily small circle which surroundsthis point.

Suppose that the single-valued analytic function wi (z) is defined on asubdomain D1 of a domain D (see fig. 5-13). And suppose that D neither con-tains any singular points of wi (z) nor is it possible to construct a closed curvewithin D which surrounds a branch point wi (z). Then it is impcissible for anytwo analytic continuations of wi (z) along paths which lie entirely within D tohave different values at any given point p of D. Now, in terms of these analyticcontinuations, we can construct on D (in the manner described above) ananalytic function w(z). Then this function will be single valued. It is an exten-sion of wi (z) from the domain D1 to the larger domain D. And the fundamentaltheorem shows that w(z) is the only single-valed analytic function with thisproperty. Therefore, when no confusion is likely to arise, we do not distin-guish between the two functions w(z) and wit (z) and we simply say that wi (z)is defined on the larger domain D.

For example, consider the function w1(z) defined by

wl (z) = z1/2

"This result is known as the monodromy theorem.

(5-117)

141


FIGURE 5-14.Analytic continuation of function zit'.

on a domain D1 which includes a portion of the real axis but which includesneither the origin nor any portion of the negative real axis, as shown in figure5-14. It is easy to verify that equation (5-17) represents a single-valued analyticfunction in the domain D1. In order to proceed it is convenient to introducethe polar representation z=rei° discussed in section 5.1. Then the function(5-17) can be written as64

wl(rei°)= ril2ei812

When the point z is in D1, the argument 0 will always lie in the range

rr <0 <Ir

(5-18)

(5-19)

(Notice that strict inequality signs are used.) Since the formula (5-18) withan extended range of 0 determines an analytic function at each finite point ofthe complex plane except r= 0 and sincethis function coincides with w1(z)in D1, we can use this formula to analytically continue wi(z) outside of D1.

" Recall that we have adopted the convention that the square root of a positive real number is always the positivesquare root.

142


Now the origin is a sing,luar point of wi (z). Hence, suppose we analyticallycontinue this function to the point p along ri, the semicircle with radius Rin the upper half plane shown in figure 5-14. Then the value of this analyticcontinuation at p is Roe"12) = R112 [cos (7112) + i sin (r/2)] = iR112. But thevalue at p of the analytic continuation along the semicircle r2 is R1lze-0.12)=R"2 [cos (-7712)+ i sin ( ir/2)] = i/02 These analytic continuations ofthe function wt (z) therefore have different values at the point p.

If, instead of stopping at the point p, we carry out the analytic continuationof w1(z) first along fi to the point p and then along "2 from the point p to thepoint q (in the direction opposite to the arrows), we arrive at the valueR112ei2v/2=_R 1/2. But since the original value of w1(z) at the point q is R112e10=R1/2, we see that the function does not return to its original value upon ana-lytic continuation around this circle. And since the radius R is arbitrary, thisshows that the origin is a branch point of w1(z). By making the transformationz= 11w and taking R arbitrarily large, we can also show that the point atinfinity is a branch point of this function. And since z =0 and z= co are theonly singular points (and therefore the only branch points) of wi (z), thisfunction will always return to its original value when it is analytically continuedaround any path which does not enclose the origin.65

Now every analytic continuation of wi (z) can be obtained from the formula(5-18) by letting r range between zero and infinity and letting 0 take on allvalues both positive and negative. But, since e2in ur=1 for n= 0, ± 1, -2.- 2, . .

we need only consider values of 0 in the range 00 0 < 00 +47r (where 00can be chosen as any fixed number) in order to obtain all possible values of thefunction (5-18). Thus, the complete analytic function iv(z) obtained fromwi (z) is the multiple-valued function defined by

17;(rew)=-- rif2ei°12 0 r 00; 0 0 = 0 . 0 0+ 4 7 r (5-20)

It takes on two distinct values at each finite point of the complex plane exceptat the origin, which is a branch point. And one of these two values is equal tothe negative of the other.

. " It will never return to its original value when analytically continued once around any path which does enclosethe origin.

143


lal

lbl

(a) Branch, cut in arbitrary location.(b) Branch cut along negative real axis.

FIGURE 5 -15. Branch cuts for z2/2.

It is usually undesirable to deal with multiple-valued functions.66 We canavoid doing this by drawing a line connecting the branch point of the functionZ112 at the origin with its branch point at infinity, as shown in figure 5-15(a),and then restricting the analytic continuations so that they are not carried outalong any path which crosses this line. Such a line is called a branch cut andit is used to prevent analytic continuations from being carried out along curveswhich encircle the origin. The actual location of the branch cut is arbitrary butwe may, for definiteness, assume that it lies along the negative real axis, asshown in figure 5-15(b). Then starting with the original function (5-17) (which,as can be seen from eqs. (5-18) and (5-19), is positive along the positive realaxis) and analytically continuing this function along all allowable paths in thecomplex plane, we obtain the extended function

"After all, we would not expect a well-defined physical problem to have a solution which is multiple valued.

144


Frit(reio) rl eicy2 0 r co; 7T 0 E 7r

which is single valued and analytic at every point of the complex plane not lyingon the negative real axis. We can also analytically continue the function whichis equal to the negative of the original function (5-17) in DI along all allowablepaths in the cut plane to obtain the extended function

W2( reo) = 0 <r<co; < 0 <377-

And this function is also single valued and analytic at every point of the complexplane not lying on the negative real axis. The functions W1 and W2 are said tobe branches of the double-valued function (5-20). Taken together these twobranches assume all the values of the multiple-valued function and are there-fore equivalent to it. Hence, we can deal with a multiple-valued function byreplacing it with its single-valued branches.

The complete function in this example is double valued and thereforehas two branches. However, we also encounter multiple-valued functionswhitth take on infinitely many values at each point and therefore have infinitelymany branches. For example, the function w(z) = In z is defined in polarnotation to be

w(rei®) = In rei° = In r+ i0

In order to obtain the complete analytic function we must (as in the precedingexample) let r take on all values in the range 0 < r < 00 and 0 all real values.But since z= reit )=-- rei(c+2") if, and only if, n= 0, ± 1, ±2, . . ., this completefunction must have infinitely many values at each point. And these valuesdiffer from one another by multiples of 27ri. This function also has a branchpoint at the origin and a branch point at infinity. And if the branch cut isagain taken along the negative real axis, the infinitely many branches wn(z)for n= 0, ± 1, -± 2, . . . of In z become

wn(rei9 =ln r+ i0 0 < r < co; (2n 1) < 0 < (2n + 1) irfor n=0, -±-1,:±2, . . .

The branch corresponding to the range 7r < 0 < 7 is called the principalbranch of the logarithm.

145


The function za (where a is some complex constant) can be defined in termsof the logarithm by the formula za = ea In Z. Hence, the branch points of thisfunction can be located only at the origin and at infinity. By using polar notationwe can express this function in the form za= ea in reai0= raeia0. And this shows

that the complete function is multiple valued unless a is an integer. In fact,it takes on infinitely many values at each point unless a is a rational number.The various branches of this function can be formed in the same way as for thelogarithm.

In a similar manner, it can be seen that for any finite point zo the completeanalytic function associated with (z zo)a is multiple valued whenever a isnot an integer. Its branch points are z= zo and z= co and its branch cut can betaken along any line joining these two points. However, once a branch cut hasbeen chosen, the various branches of this function will then be analytic every-where in the cut plane.

5.8 PERMANENCE OF FUNCTIONAL RELATIONS

Let F(zi, . . zn) be a complex function of the n complex variableszi = xi + iyi, . zn=ocn+ iy.. If this function can be expanded in a powerseries

F(zi, . . zn) = . . ., ainzi142 . . 4n. , in=i

with complex coefficients ail, . . ain and if this series converges in someneighborhood of each point in some domain of the 2n-dimensional space whosecoordinates are xi, . . xn, yi, . . yn, then we say that F is an analyticfunction of the n complex variables zi, . . zn in D. This is clearly an exten-sion of the definition of an analytic function of a single complex variable givenin section 5.2.

Now let F (wi , . . wn, z) be an analytic function of the n +1 complexvariables wi, . . wn, z for all values of the variables w1, . . wn and forall values of z in some domain Do. If wi (z), . . wn(z) are analytic functionsof the complex variable z (in the usual sense) in some common subdomainD of Do, then it can be shown (ref. 4) that the function g(z) defined by

146


--Path of analyticcontinuation ofthe functionswi(z) , wn(z)

FIGURE 5-16.Illustration of permanence of functional identities.

g(z) F(wi(z), ,wn(z),z)

is also analytic in D.Now suppose that /hi (z), . . run (z) are analytic continuations of

wi (z), . . wn(z), respectively, from D to some other common subdomainD of Do and these analytic continuations can all be obtained by analyticallycontinuing the functions wi (z) , . . wfl(z) along a single curve F whichlies entirely within Do. (This is illustrated in rig. 5-16.) In addition, supposethat wi (z); . ., W2 (Z) satisfy the equation

F(wi, . . .,wn,z)=0 (5-21)

at all points z in D. Then by analytically continuing these functions along F toD and using the fundamental theorem of analytic continuation given in thebeginning of section 5.7, it can be shown (ref. 19, p. 210) that ivi(z), . .

fun (z) also satisfy equation (5-21) at every point of D. This is known as theprinciple of permanence of functional relations. Roughly speaking, it meansthat the analytic continuations of the solutions of equation (5-21) are alsosolutions of this equation.

147


5.9 DIFFERENTIAL EQUATIONS IN COMPLEX PLANE

5.9.1 Definition

We have already indicated the utility of extending the definition of adifferential equation to include the case where the variables are complex. Tothis end let F (wi, . . . , wn, z) be an analytic function of the n+ 1 complex vari-ables w1, . . wn, z in some domain D. Then the nth-order normal differentialequation in the complex domain is an equation of the form

dnw dw. dn'tww' 7i' ' z (5-22)

Notice that in writing this equation we imply that its solutions, if they exist,must possess complex fietivatives at all points where they are defined: It isalso reasonable to require that the solutions satisfy the equation at least on somedomain in the z-plane. Hence, the solutions to equation (5-22) must be analyticfunctions. This is a much stronger restriction than is imposed in the case of realvariables, where we require only that the solutions be sufficiently differentiable.

5.9.2 Fundamental Theorem

The following fundamental theorem (which is analogous to that given inchapter 1 for the real-variable case) can be shown to hold (ref. 4, p. 119). Foreach point 67 61 ., n, zo of the domain D where the function F is analytic,there exists a unique (i.e., single-valued) function w(z) which satisfies the initialconditions

w (zo) = rl, W' (zo) 01-1)(ZO) = Co

is analytic, and satisfies equation (5-22) in some neighborhood of the point zo.We shall be principally interested in the (effectively normal) linear equation

d"w dn-+ ai(z) dznw + . . . + an_1(z)

dwF an(z)w+ b(z)= 0 (5-23)

dan

where the coefficients ai (z) 9 an (Z) b (z) are all analytic on some commotdomain Do. This equation is effectively of the form (5-22) with the function F

67 41, zo are n+ I complex

148

P.EVIEW OF COMPLEX VARIABLES

analytic on the domain D which consists of all the values of the variableswt, . . w2 and all the values of the variable z which lie in the domain Do.Hence, the fundamental theorem now becomes: For each set of complexnumbers 6, . . 4, and each point zo of Do there exists a unique functionw(z) which satisfies the initial conditions

wi(z0).42, w(nI)(zo) = 4n

is analytic, and satisfies equation (5-23) in some neighborhood of the point zo.There is also, in this case, an additional result, due to Fuchs (ref. 24, p. 4)

which asserts that this solution, w(z), has a Taylor series expansion about zowhose radius of convergence is at least equal to the shortest distance betweenzo and the boundary of Do.

Now suppose that w(z) is a solution to equation (5-23) on some subdomainDI of Do. Then the (single valued) function w(z) and all its derivatives areanalytic on DI. Let iii(z) be an analytic continuation of w(z) along some curvein Do to some other subdomain D of Do, as shown in figure 5-17. Then since,as indicated in section 5.7.2, the analytic continuation of a derivative of ananalytic function is equal to the derivative of the analytic continuation, we canapply the principle of permanence of functional relations to equation (5-23)

Domain of analyticityof coefficients of dif-ferential equations

Path of analyticcontinuation ofw(z), w'(z)

FIGURE 517.Illustration of analytic continuation of solution to differential equation.

149


to show that the analytic continuation w(z) is itself a solution to equation(5-23). Thus,, the analytic continuation along any curve lying entirely withinDo of a solution to equation (5-23) is also a solution of this equation.


has the solution

1 1

w" w' +Kilv=°

w(z) =e12+

But upon analytic continuation of this solution around any closed path en-circling the origin, we obtain the function

w1 (z) =(e2riz)1/2 (e2iriz )1/3 = z1/2 e2rrif3z1/3

And it is easy to verify by direct substitution that wl (z) is also a solution of theequation. (In fact, w(z) and wi (z) are linearly independent.)

We shall now show that any solution to equation (5-23) which is definedon a subdomain of Do can indeed be analytically continued along any carve inDo. This means that no solution to equation (5-23) card, have a singular pointin Do. The assertion can be proved by assuming that there exists a solutionw(z) in a subdomain D1 of Do which cannot analytically continued alongsome curve F in Do and then showing that this leads to a contradiction. Thus,if w(z) cannot be continued along F, there must be a point zi on F as shown infigure 5-18 such that w(z) can be continued up to, but not past, this point. Wecan therefore choose a point z2 on the portion of F joining D1 to zt which is closerto z1 than any part of the boundary of Do. Upon analytically continuing thesolution w(z) along F to z2, we obtain a solution w2 (z) of equation (5-23)in a neighborho;-1 of z2. And Fuchs' result shows that w2(z) can be expandedin a Taylor series about 22 that converges in a circle which includes the pointz1. But this constitutes an analytic continuation of w(z) along F past the pointzi, which was assumed to be impossible. And this proves the assertion.

It follows from these results that any solution 68 to equation (5-23) in

"The existence of such a solution is asserted by the fundamental theorem but only in some neighborhood of thispoint. And this neighborhood could be very small.

150

REVIEW OF COMPLEX VARIABLES .

tCircle of convergence ofTaylor series expansionof w21z)

FIGURE 5-18.Continuation of w(z) along 1'.

a neighborhood of a point where the coefficients are analytic can actually beextended to obtain a solution to this equation on the entire domain Do on whichthe coefficients are analytic. However, in order to do this we must, in general,allow these solutions to be multiple valued.

5.9.3 Linearly Independent Solutions

By using the fondameni al theorem given in this section, the various resultsgiven in section 1.6 for linear equations with real variables can be extended tothe complex-variable case. Thus, the homogeneous equation associated withequation (5-23)

dnw do -lw+ ai(z)+ . . . -1-an_1(z)w= 0dz" dzn--1

(5-24)

possesses n linearly independent (single valued) solutions w1 (z), ., wn(z),called a fundamental set of solutions, in a neighborhood of each point ofDo. Now let wp(z) be a particular (single valued) solution of equation (5-23)in the neighborhood of some point of Do. Then every solution of this equationabout this point can be obtained by making a suitable choice of the arbitraryconstants c1, . c7, in the general solution w(z) = (z) + . . . +c,,w,(z)+wp(z) of this equation.

151


The definition and discussion of linear independence given in section 1.6applies with only trivial modification to the case where the functions are com-plex and analytic. Thus, in particular, a necessary and sufficient condition(ref. 24, P. 13) that n (single valued) solutions w1(z), . . wo(z) of equation(5-24) be linearly independent on a domain Do on which the coefficients areanalytic is that the Wronskian

IV( 14'1 Wn )

not be equal to zero at any point

W2

W2

win-1) W(")2

zi of Do. Since wi, . .

Wn

14)n

(n-1)wit

:co are analytic

(5-25)

func-tions of z, the Wronskian itself is an analytic function of z which we shall denotebyV(z). Thus,

IV(z) = 1V(w1(z), . . wo(z))

It is also easy to see that the Wronskian of the analytic continuations ofw1 (z) , . . . , w,(z) along a curve F in Do is equal to the analytic continuation ofthe function r(z) along r.

It can be shown by using the rules for differentiating determinants and bysubstituting in the differential equation (5-24) (ref. 24, p. 12) that 7r satisfiesthe first-order differential equation

= (z) 7p, (5-26)

And upon separating the variables and integrating along any path I' in Do,we obtain

7V (z) = ce (5-27)

where c is a complex constant of integration.

152


1Since pi (z) is analytic in Do, its integral pi(z)dsz must also be analytic

in Do. Hence, in particular, this integral cannot become infinite at any point ofDo; and therefore the exponential factor in equation (5-27) can never vanishin Do. Thus, 71r(z) can only be equal to zero at a point zi of Do if c =0. Andthis shows that, iflir(z) vanishes at any point of Do, it must vanish at every pointof Do.

In the special case where n =2, equation (5-27) can be used to obtain anexplicit formula which determines a second linearly independent solutionW2(z) to equation (5-24) when one solution wi(z) to this equation is known.Thus, when n =2, we find upon expanding the determinant and rearrangingThat

d w2

dz)=7r(z)=ce-f Awd,w1

And integrating this along any curve in Do yields the formula

-f p.wd.w2(z) =-- cwi (z) J dz

(z) J2 (5-28)

which agrees with the formula obtained by the method of variation of parame-ters in section 4.1. It follows from the way in which it was constructed that anysolution w2(z) calcOated from this formula will be linearly independent of

(z).More generally, it can be shown (ref. 24, p. 16, example 10) that, if n-1

linearly independent solutions, say wi(z), . . wn_1(z), to equation (5-24)are known, another linearly independent solution to this equation is given by

n-1

Pe+d-i)ch

illi(z)dzra )Zi2Wn=C wt(z)

t=1

where r, z} is the Wronskian

153


Wi(z) = W(wi, .

WI W2 Wm 1W W1 2 n I

win-2) win 2) w(n 21n

and, for i = 1, 2, . . n 1, M i(z) is the cofactor of u.1-2 in this Wronskian.That is,

awM,(z) awr1

5.10 NONELEMENTARY TRANSCENDENTAL FUNCTIONS

We shall have occasion to use two particular nonelementary functionsof a complex variable called the gamma function and the beta function. First,we define the analytic function I'(z) for.We z > 0 to be the Eulerian integralof the second kind.

r(z) (5-29)

This integral converges in the right half plane z > 0 and diverges forz 0. It can be shown (ref. 25) that it represents an analytic function in its

domain of convergence. Although this analytic function is only defined byequation (5-29) in the right half plane, it can be analytically continued into theleft half plane .0, z 0 by using the formula

1.(z)r 7r(1 z) = (5-30)

sin TTZ

to compute the values of 1r(z) for z < 0 from its values at points in theright half plane. It follows from this equation that 1-(z) has simple poles atz= 0, 1, 2, . . .

Integrating equation (5-29) by parts (with z replaced by z + 1) showsthat for A. z > 0

154

Hence,


r(z + 1) = z e-ttz-idt0

r (z + 1) = zr(z) (5-31)

By successively applying equation (5-31) we find that for any positiveinteger n

r(z + n) = (z + n 1)r(z+ n 1)

= (z + n 1)(z + n 2)r(z + n 2)

=(z+n-1)(z+n 2) . .

It is convenient to introduce a special notationr(z) in the last member of this equation. Hence,factorial function (z). by

and

n -1

n (.';; + m)m=o

= z(z + 1)(z + 2)

(z) 0 = 1

. (z + 1)zr(z) (5-32)

for the factor multiplyingwe define the generalized

(z + n 1) for n = 1, 2, 3, . .

(5-33)

Thus, the symbol (z), denotes the product of n factors, each factor beingone larger than the preceding one: For example,

(7)3= 7 x 8 x 9

(1) _1x3xS 7

' 488-942 0 - 73 - 11155


Notice that when z = 1 in equation (5-33), we obtain the ordinary factorialfunction since

W.= 1 2 .3 ... n = n! (5-34)

By using this notation, equation (5-32) can be rewritten as f(z+ n) = (z),I.(z).And, therefore, the generalized factorial function can be expressed in termsof the gamma function by

f(z + n)(z). for n = 1, 2, 3, . . (5-35)r(z)

Since integrating equation (5-29) with z = 1 shows that I'M = 1, we findfrom equations (5-34) and (5-35) that

r(n +1) = n! for n = 1, 2, 3, . . (5-36)

This equation is also valid for n = 0 provided we use the usual definition 0! =1.The beta function B(z, C) is a function of two complex variables and is

defined by

B(z, C) = tz-1(1 t)c-idt0

(5-37)

for .0, z > 0 and 4.9: > 0. Howevet, ii ib possible to express this functionin terms of the gamma function since

r(z)1' = (1 e-Itz-idt)Joy

e-Irt-ldr)

= e-itz-1(1 e-r7c-Idr) dtJ 0

And upon setting x = 7/t, we can eliminate 7 from this equation to obtain

r(z)r(o e-ttz-lit (r e-txxc-idx) dt0

156


Hence, after changing the order of integration we find that

rx.

r(z)r(4) = , xt-i (1 e-tu-4-otz+c-Idt dxJo o

Then by defining n by Ti = t(x + 1) and eliminating t, we getnx-iz)noJo

= (1 +tx)c+ z

dxJo

xt--1= r(z + C) fo(1 +x)c+2

And after setting x = t/(1 t) this becomes

dx

r(z) no = r (z+ 4) f tt--1 _oz-ith.

But comparing this with equation (5-32) shows that

B _r(x)r(4)(z+ 4)

(5-38)

CHAPTER 6

Solution of Linear Second-Order Differential

Equations in the Complex Plane

In section 5.9 we extended the definition of a differential equation toinclude the case where the variables are complex. In this chapter the ideaspresented in that section are used to find the behavior of and, in certaincases, to construct solutions to the second-order differential equation

dd2 zw2 + P (z) + (z) w = 0 (6-1)

where z is a complex variable and the coefficients p(z) and q(z) are analyticfunctions in some domain D of the z-plane. We have seen that every solutionof this equation is an analytic function of z.

A point at which the coefficients of equation (6-1) are both analytic iscalled an ordinary point of this equation. Thus, every point of D is an ordinarypoint. A point zo which is a singular point, of either p(z) or q(z) (or both) issaid to be a singular point of equation (6-1). And if the only singularities ofp(z) and q(z) which occur at zo are isolated singular points, zo is also saidto be an isolated singular point of equation (6-1).

6.1 GENERAL BEHAVIOR OF SOLUTIONS AT ORDINARY POINTS

Let zi be any point of D. It was indicated in section 5.9 that equation(6-1) will possess two linearly independent (single valued) solutions, wi (z)and w2(z), in some neighborhood of z1. These solutions are said to be a funda-mental set of solutions. They possess Taylor series expansions about z1, say

"Notice that this definition is consistent with the one given in section 1.5.

159


wi(z) = anz zi)nn=0

w2(z) = an'-) (z zi) nn=9

(6-2)

whose radii of convergence are at least equal to the shortest distance betweenzi and the boundary of D. Since the coefficients a' equation (6-1) can alwaysbe analytically continued across those points on the boundary of D whichare not singular points of the equation, it is always possible to extend thedomain D (in which the coefficients are analytic) in such a way that everyfinite point on its boundary" is a singular point of equation (6-1). Hence, wecan assert that the radii of convergence of the Taylor series expansions (6-2)of the solutions w 1(z) and w2(z) are at least equal to the distance Ro betweenz1 and the nearest singular point of equation (6-1). It was also shown insection 5.9 that the solutions wt (z) and w2 (z) can be analytically continuedalong any simple curve in D and that these analytic continuations are them-selves solutions of equation (6-1). Thus, in particular, we can assert thatthe series (6-2) not only converge within a circle of radius Ro but that theyalso satisfy the differential equation everywhere within this circle.

6.2 GENERAL. BEHAVIOR OF SOLUTIONS NEAR ISOLATED SINGULAR POINT

Suppose that zo is either an isolated singular point or an ordinary pointof equation (6-1). Then the coefficients p(z) and q(z) of this equation will beanalytic within the punctured circular domain 0 < z zo I < R (shown infig. 6-1) whose radius R is equal to the distance between zo and the singularpoint 71 of equation (6-1) closest to zo. We shall call this domain 72 A. Nowlet wi (z) and w2(z) be a fundamental set of solutions to equation (6-1) ina neighborhood Do of a point z of A. We have seen in section 5.9 that thesesolutions can be extended (by using their analytic continuations) so thatthey satisfy equation (6-1) at every point of A. However, since zo may bea singular point and since it can be encircled by a curve such as the curver shown in figure 6-1, these extended solutions can be multiple valued.

70 Provided D has a boundary." Other than zo itself."Notice that zo is not a point of A.

160

SECOND-ORDER EQUATIONS IN THE COMPLEX PLANE

FIGURE 6-1. Contour for analytic continuation of wi and w2.

In order to deduce the behavior of these extended solutions, we shallnow construct a fundamental set of solutions on A whose structure is particu-larly transparent. To this end let the fundamental set w1 (z) and w2(z) beanalytically continued from Do counterclockwise around F. This process will,in general, yield two new functions of z, say W1 (z) and W2 (z) which are definedon Do by

WI (z) = wi (e2Tri (zzo) +zo)

W2 (z) **-= w2,(e2vi (z zo) +zo)(6-3)

But we know that W1 (z) and W2 (z) must satisfy the differential equation (6-1)and that every solution to this equation in Do can be expressed as a linearcombination of the fundamental set of solutions wl (z) and w2(z). Thus, thereexist complex constants. a11, a12, a21, a22 such that

(z) = aitwi(z) +a12w2(z) 1

TV2 (z) = avw, (z) + a 22w2 (z)(6-4)

161


For example, consider the differential equation 73

,w +6z26z 6z2

and let zo= 0. A fundamental set of solutions to this equation is wi=z1/2 andw2=z113. Upon applying equations (6-3) to these solutions, we find that

and

fri1 (z)=w1(e2iriz)= (e iz)1/2= ( 1)w1 (z)

W2 (z) = e2lri/321/3= e2lri/3W2(Z)

And, therefore, the constants which appear in equations (6-4) become, in thiscas_ e a11=-1, a12=0, a21=0, a22=e217(13.

We shall now show (in the general case) how the constants in equations(6-4) can be used to construct the desired fundamental set of solutions toequation (6-1) on the domain A. In order to do this, we first recall from thetheory of linear equations that the algebraic equations

(an X)ci + anc2 = 0

anci + (a22 X)c2 = 0(6-5)

have a nontrivial solution74 in ci, c2, provided X is a root of the characteristicequation

an X a21

d12 a22 A = X2 ( + a22 ) X + ( an a22 a nazi ) = 0 (6-6)

This equation will either have two distinct roots or two equal roots. We shalltreat these two cases separately.

vs This equation was discussed in the example given in section 5.9.2.74 That is, a solution other than c1 = cz = 0.

162


6.2.1 Case I: Roots of Characteristic Equation Distinct

First, suppose that the zwo roots of equation (6-6), Xi and X2, are distinct.Let cci), 41) be a set of nontrivial solutions to equations (6-5) correspondingto the root X1 and c12), cP be a set of nontrivial solutions corresponding tothe root X2. Thus,

and

(all Xl)cco + a2141) = 0

ancco + (a22 X1)40 = 0

(an X2)62) + a2,42) = 0

a12c12) + (a22 X2)c(2) = 0

(6-7a)

(6-7b)

Now let u1(z) and u2(z) be the solutions to equation (6-1) which aredefined in Do (and only in Do) in terms of the fundamental set w1(z) andw2(z) by

u1 (z) = cV)wi (z) + cV)w2(z) (6-8)

u2 (z) = cPwi (z) + cVw2 (z) (6-9)

Then it follows from equations (6-3), (6-4), and (6-8) that ui(e2/4(zzo)+zo),the analytic continuation of u1(z) counterclockwise around F, is given by

ui(e2771(z zo) + zo) = c(1)w1(e27ri(z zo) + zo) + cV)w2(e2lri(z zo) + zo)

= c4' + c4')W2 (z)

= + cV)a21] (z) + [c1 oal2 + cV)a22]w2 (z)

Hence, equations (6-7a) and (6-6) imply

ui(e27ri(z zo) + zo) = Xiccouh(z) + X1cV)w2(z)=X1u1(z) (6-10)

163


And the same argument shows that the analytic continuation of the solutionu2(z) around u2(e27ri(z zo) + zo) , is

u2 (e2lri(z zo) + zo) = X2u2 (z) (6-11)

The constant X1 is nonzero. If this were not true, equation (6-10) wouldshow that ui (e2ri(z zo) + zo) is identically zero in Do. But then ui(el'Iri(z-zo)+ zo) could be analytically continued backwards along F to show that -ui (z)is identically zero. And since the constants eio and cc)) are not both zero,equation (6-8) would show that wi (z) and w2 (z) are linearly dependent. Butthis is impossible since wi(z) and w2(z) are, by hypothesis, a fundamentalset. Hence, we conclude that X1 0 0. The same reasoning shows that A2 0 0.We can, therefore, define two (finite) numbers 61 and 62 by 75

SI27rt

In X162 2=

In A2 (6-12)

and use these numbers to define the two analytic functionsfi (z) andf2 (z) inDo by

fi (z) = (zzo) 81ui (z)(6-13)

f2 (z) = (z zo) 8 2u2 (z)

Since the right-hand members of these equations are products of functionswhich can be analytically continued along any simple curve in A, it follows that.11(z) and f2(z) must also have this property. And since wi (z) , w2(z), (z-zo)''and (z-zo)-82 have no singular points in A, it follows from equations (6-8),(6-9), and (6-13) that f (z) and f2 (z) have no singular points in this region."Hence, these functions have no branch points in A. On the other hand, theanalytic continuations fi(e2n1(z-zo)+ zo) of fi(z) for i=1, 2 in a counterclock-wise direction around any cn've I' which encloses zo are

fi(e2ni(z- zo) +zo) = (e2ni(z- zo)) iti(e2fri(Z ZO) +zo) for i=1, 2

75 The condition X, 0 Xo implies that 8 e is not an integer.76 However, the point zo may be a singular point of these functions.

164


Hence, it follows from equations (6-10) and (6-11) to (6-13) that

1fi(e"i(z- zo) +zo) =At

(z- zo)-6iut(e"i(z- zo) +zo)

= (z- zo)-8iui(z)=fi(z) for i= 1, 2

But this shows that the point zo is also not a branch point of the functionsfi(z), 1, 2. Hence, these functions will return to their original values whenthey are analytically continued around any closed curve in A. And this implies,as indicated in section 5.7, that they can be extended to single-valued analyticfunctions on the entire domain A. We can therefore suppose that these ex-tensions have already been carried out and that fi (z) and f2(z) are defined onthe entire domain A. Hence, it follows from equations (6-13) that the extensionsof the solutions ui(z) and u2(z) from Do to the domain A (which we shallalso denote by /II (z) and u2(z)) are functions of the form

ui (z) = (z zo)811-1(z)

u2(z) = (z-zg)82f2(z)(6-14)

where fi (z) and f2(z) are single-valued analytic functions on the entire domainA. (The point zo, however, may be an isolated singular point of these func-tions.) And as indicated in section 5.8 the extended functions ui (z) and u2(z)must satisfy the differential equation (6-1) everywhere within A. These arethe solutions which we have set out to construct. We shall now show thatthey are linearly independent. To th end suppose that yi and 72 are anytwo constants such that

yiu, (z) + y2u2(z) = 0

Substituting equations (6-8) and (6-9) into this equation shows that

[ + y2c;2) J w1(z) + [ y1c(21) + y2622) I w2(z) =0

And since w1(z) and w2 (z) are linearly independent, it follows that

165


yicV) + y2cc2) = 0

vic(21) 4. ,y2c(22) =0(6-15)

Hence, upon multiplying equation (6-7a) by yi and equation (6-7b) by y2and adding the results, we find that

(A1 x2)yici.1) = (A1 x2)72cv = 0

But since we are considering the case where Al 0 A2, this implies that

yicco= y2cV) = y2cc2) = yicP= 0

And by using the facts that cc') and 41) are not both zero and c?) and cV) arenot both zero, we can conclude that y, = y2 = 0. But this shows that ul (z)and u2(z) are linearly independent.

Before discussing the implications of these solutions, we shall first treatthe case where the roots of the characteristic equation are equal.

6.2.2 Case II: Roots of Characteristic Equation Equal

Thus, suppose that the roots of equation (6-6) are equal. Then the argumentused in the preceding section can easily be adapted to show that there stillexists at least one solution to equation (6-1) of the form

wt (z) = (z zo) (z) (6-16)

where ft (z) is a single-valued analytic function in z and 81= (1/27ri) In Alwith Al 0 0. We can therefore assume that the fundamental set if solutionswi (z) and w2(z) of equation (6-1) in the domain Do (fig. 6-1) has been chosenin such a way that w, (z) is given by equation (6-16). By analytically con-tinuing this solution around r we find that in this case the function W1(z)(defined in eqs. (6-3)) is given by Wt (z) = kiwi (z). And since wi (z) and w2(z)are linearly independent, the first equation (6-4) shows that

166

and


a12 = 0

all =

However, these equations imply that the roots of the characteristic equation(6-6) are X1= all 0 0 and X2= a22. But since we are considering the casewhere Xi= X2, it follows that an= a22= Xi. Hence, in this case equations(6-4) become

(z) = X au' (z)

TV2(z)=a2iwi (z)+Xiw2(z)

We now define the function 12(z) on the domain Do by

f2 (z) = (z zo) -8.w2(z)a21fi (z) ln (z-zo)

27riXt

(6-17)

(6-18)

This function can be analytically continued along any simple curve in A, andit will return to its original value when it is analytically continued along anyclosed curve which does not encircle zo. Its analytic continuation (counter-clockwise) along any curve r which encircles zo is

f2(e27ri (z zo) + zo) = (z -zo) -81e-27'184V2 (z)a21f1 (z)

In e27"(z zo)2riXi

= (z -x"i ) W2(z)2ifi(z) In (z-zo) a21

(z)

But inserting equation (6-16) and the second equation (6-17) into this relationshows that

167


f2 (e21ri(z zo) + zo) =f2 (z)

Hence, the function f2(z) will return to its original value when it is analyticallycontinued around any closed curve in A. It can therefore be extended to asingle - valued analytic function on the entire domain (which may have anisolated singular point at zo). And we can now conclude from equations (6-16)and (6-18) that the extension of the solution w2(z) from Do to the entire domainA (which we shall also denote by w2(z) ) is a function of the form

w2(z) =-- (zzo)V2(z) +awi(z) In (zzo) (6-19)

where f2(z) is a single-valued analytic function on the entire domain A and wehave put a = a21/27riX1. It can again be shown that the solutions w1(z) andw2(z) are linearly independent.

6.2.3 General Conclusions

Notice that the fundamental set of solutions (6-16) and (6-19) with a =0are of the same form as the fundamental set (6-14) qith S1= 62. The funda-mental set (6-2) which occurs at an ordinary point of equation (6-1) is a specialcase of the fundamental set (6-16) and (6-19). We have therefore establishedthe following conclusion: Let zo be an ordinary point or an isolated singularpoint of equation (6-1) and let R be the distance between zo and the singu-lar point of equation (6-1) which is closest to zo. Then equation (6-1) possessesa fundamental set of solutions on the punctured circular region 0 < z zo1 < Rwhich is of the form

(z) = (z 2'0V, (z)(6-20)

wz (z) = (z zo)82/2(z) + awl (z) In (z zo)

where f1(z) and f2(z) are analytic single-valued functions in 0 < z zo <and a, 61, and 62 are complex constants.

The fundamental set (6-20) is called a canonical basis. The constant ais equal to zero whenever 81 52 0 0, ± 1, -± 2, . . .. The case where 81 62= 0, ± 1, ±2, . . . is referred to as the exceptional case. Notice that theconstant a may also vanish in the exceptional case. This occurs, for example,when zo is an ordinary point of equation (6-1).

168


We should not conclude from these results, however, that the Solutionsof equation (6-1) will always have singularities at the singular points of thisequation. For example, the equation

z2w" 2zw' + 2w = 0

has an isolated singular point at z = 0. But the general solution to this equationis

w = A oz ± A iz2

where Ao and A1 are arbitrary. And this shows that every solution of thisequation is analytic at z= 0.

By changing notation we can rewrite this solution in the form

w = ao al (z 1) + (al ao)(z 1)2

where ao and al can now be taken as arbitrary. This is evidently a canonicalbasis at the ordinary point z = 1. Since it is an entire function, it certainlyexists and satisfies the equation within a circle whose radius is larger thanthe distance between z =1 and the nearest singularity (namely, z= 0) of thedifferential equation.

If either of the functions fi (z) :1Ladj2(z) in equations (6-20) is not analyticat zo, then zo must be an isolated singular point of this function. Hence, fi (z)and 12(z) can always be expanded in the Laurent series 77

fi(z)= mp(zZo)nn= OD

(6-21)

f2(Z) (Z Zo)nn= 00

which converge for 0 < lzzoi < R. An important special case occurs whenthe functions fl (z) and f2(z) either have poles or are analytic at zo. Then theseries (6-21) contain at most a finite number of negative powers of z zo andcan therefore be written in the form

77 Which may or may not contain negative powers of (zzo).

169


ft. : blp(zzo)nn=ri

for i= 1,2

where r1 and r2 are finite integers (they may be negative or zero). However,we can shift the index in the sums by vatting k= n+ ri and then summing onk. Thus,

ii(z) ( ___ zIkri = (ZZ0)ri a")(zZo)kv/k=0 k=0

for i= 1, 2

where we have put ak1= M:lri for i =1, 2.Inserting these equations into equations (6-20) shows that in this case the

canonical basis is of the form

IUI (z) = (z zo)P. E ao)(z zo)kk=0

(6-22)

74)2(z) = (z zo)P2 a(2)(z zo)k + awi(z) In (zzo)k=0

where we have put pi= 51 r1 and p2 = r2. The series converge everywherewithin the circle jz < R (including the point zo) and they, therefore, repre-sent analytic functions within this circle. The solutions WI and w2 are said tobe regular. It follows from the fact that r1 and r2 are integers that a vanisheswhenever pi N00,7_171,71-2, . . . since it vanishes when Si 82 has thisproperty.

We have already seen that equation (6-1) will always possess two regularsolutions (with p1= p2 = a= 0) at the point zo whenever zo is an ordinary pointof this equation (i.e., if p(z) and q(z) are analytic at zo). We shall now deter-mine certain conditions which the coefficients of equation (6-1) must satisfyat an isolated singular point zo if this equation is to possess two regular solutionsabout zo.

Since wi (z) and w2(z) are linearly independent in 0 < Iz zol < R, theirWronskian (see section 5.9.3)

7fr (z) ---=

170

w2 (z)

to; (z)

(6-23)

SECOND-ORDER al:MATIONS IN THE COMPLEX PLANE

cannot vanish at any point z of this domain. On the other hand, putting00

gi E al.1)(z zo)k for i = 1, 2 in equations (6-22) and substituting thek=0

result into equation (6-23) shows that

$r(z) = (z zo)P1 +P2-1G1(z) + a(z zo)2P r''G2(z) (6-24)

where we have set G1 = (p2 pi)g1g2 + (z zo), (gig; g;g2) and 62 gi.Since g1 and g2 are single-valued analytic functions in the entire circlelz zol < R (including zo), it is clear that Gi and G2 are also functions of thistype. And since a = 0 whenever the exponents pl + p2 -1 and 2/31 1 in equa-tion (6-24) do not differ from one another by an integer, this equation canalso be written as

Yr(z) = (z zo)AGo(z)

where X is the smaller of the two numbers pi + p2 1 and 2pi - 1 and Go(z)is a single-valued analytic function in the entire circle jz zo) < R. Hence,Go(z) can be represented by a power series

Go(z) = Ecc An(z ao)nn=0

whose radius of convergence is R. But it follows from the fact that 7r(z) isnot equal to zero in 0 < zol < R that there must be a smallest integer,say m, such that A. 0 0. Hence,

where

7V(z) = (z zo)"G(z) (6-25)

G(z) = E Ak +m(Z zo)k = (z zo-mco(z)k=0

is also a single-valued analytic function in lz zol < R but not equal to zeroat zo. And since IV(z) does not vanish at any point of the punctured region

171

488-942 0 - 73 - 12


0 < Iz zol < R, we conclude from equation (6-25) that G(z) is not equal tozero anywhere within the circle lz zot < R (including the point zo). Hence,differentiating (6-25) shows tbat

where

1 cur dlnYr X + mdz dz z zo

G' (z)G(z)

P(z)

is a single-valued analytic function in Iz zol < R since its denominator doesnot vanish in this domain. But equation (5-26) shows that the coefficientp(z) of w' in equation (6-1) is related to the Wronskian by

1 coP(z)

riTr"

Thus, if equation (6-1) possesses two regular solutions, its coefficient p(z)must be of the form

P(z)X+ m +P(z)

zo(6-26)

Similarly, it can be shown by substituting the first equation (6-22) andequation (6-26) into equation (6-1) that this equation will possess two regularsolutions at zo only if the coefficient q(z) is of the form

q(z) = , P-- _t_kz zor _ +Q(z)

where a and f3 are constants and Q (z) is analytic a zo. We have thereforeshown that equation (6-1) can possess two regular solutions at the point zoonly if its coefficient p(z) has, at most, a simple pole and its coefficient q(z) hasat most a pole of order 2. We therefore say that a singular point zo of equation(6-1) is a regular singular point if

772


..kr1) p(z) is either analytic at zo or has a simple pole at zo(2) q(z) is either analytic at zo, has a simple pole at zo, or else has a pole

of order 2 at zo.Notice, however, that we have not yet shown that equation (6-1) always

possest,es two regular solutions at any regular singular point.An isolated singular point of equation (6-1) which is not a regular singular

point is called an irregular singular point.In order to treat the point z=00, we must, as in section 5.3, introduce the

new independent variable C= 1/z into equation (6-1). Then the point z=0)will be an ordinary point, a regular singular point, or an irregular singular pointif the point C=0 is, respectively, an ordinary point, a regular singular point, oran irregular singular point of the transformed equation. But the change ofvariable C= 1/z transforms equation (6-1) into the equation

where

d2w dw+POW -2Z+Q0(4)W=0

Po(C)

Qo() 7-= q z4q(z)

(6-27)

(6-28)

(6-29)

Thus z =00 is an ordinary point of equation (6-1) if both Po(C) and 120(C)are analytic at C=0, which is equivalent to saying that 2zz2p(z) and eq(z)are analytic at zcz00. Similarly, the point z= 00 is a regular singular point ofequation (6-1) if Po(C) has, at most, a simple pole at 4= and Q0(C) has, atmost, a pole of order 2 at C=0, which is equivalent to saying that 2z z2p(z)has, at most, a simple pole at z= 00 and z4q(z) has, at most, a pole of order 2 atz= 00. If the point z= 00 is an isolated singular point of equation (6-1) but is nota regular singular point, it is an irregular singular point. It is clear that thecoefficients p(z) and q(71 themselves must certainly be analytic at z = co when-ever this point is a regular singular point.

n,ese definitions are best clarified by considering an example. Thus, thedrdfferential equation

z2(z -1)w" (z+ 1)w' +zw=0

173


can be written as

Hence,

(Z+1) 1wifZ2(z .-1)w z( z -1)

(z+1) 1P(z) .z2(z 1) q(z) =z(z -1)

And the function p has a simple pole at z =1 and a pole of order 2 at the pointz =0. The paint z =0 must, therefore, be an irregular singular pint. The func-tion q has simple poles at z= 0 and z= 1. Hence, the point z =1 is a regularsingular point. All other finite points are ordinary points. In order to considerthe point at infinity we must consider the functions 2z z2p(z) and z4q (z)(instead of the coefficients p and q) which, in this case, become

2z z2p = 2z+ z+ 1z-1

4z q z 1Z3

Hence, 2z z2p has a simple pole and z4q has a pole of order 2. And this showsthat the point at 00 is a regular singular point.

We shall now show by actually giving a procedure for finding the solutionsthat the differential equation (6-1) always possesses two regular solutions at aregular singular point.

6.3 SOLUTION OP EQUATION ABOUT ORDINARY POINTS AND REGULARSINGULAR POINTS

Let zo be either an ordinary point or a regular singular point of the differ-

ential equation

to" + p(z)w' + q(z)w=0 (6-30)

Since p has, at most, a simple pole and q has, at most, a pole of order 2 at zo,the Laurent series expansion for p about zo contains, at most, one negative

mpower of z zo and that for q contains, at most, two. Let R1 and R2 be the

174

SECONDORDER EQUATIONS IN THE COMPLEX PLANE

distances between zo and the nearest singular points of p(z) and q(z), re-spectively, to zo. Then R1 and R. will be strictly positive and

p(z)= E pk(z zo)k-' for 0 < lz zo I <R1 (6-31)k=0

,(z). E qk(z_zo)k-2 for 0 < lz zol < R2 . (6-32)k=0

The point zo will be an ordinary point of the d'ilerential equation if, and only if,

(6-33)

6.3.1. First Regular Solution

We shall now show that equation (6-30) always has at least one solutionof the form

w(z) = (z z Ean(z Zo)n1=0

(6-34)

in some punctured circular region about the point zo. This will be accomplishedby first showing that the constants p and an can always be determinedin such a way that equation (6-34) formally satisfies the differential equation.It will then be shown, a posteriori, that the formal operations were indeedjustified and therefore that the formal solution is, in fact, a true solution ofthe differential equation. The procedure which we develop by this process canthen always be. used to construct a solution to any given differential equationof the form (6-30) in the neighborhood of any of its ordinary points or regularsingular points.

Since the assumed expansion (6-34) must certainly have a leading termand since the exponent p is not, as yet, specified, we can always assumethat matters are arranged so that ao 0 0. We now substitute this expansion intothe differential equation (6-30) and suppose that the series can be differen-tiated term by term. Then

175


oz

(n + p)(n + p 1)an(z zo)n+P-2n=o

+[±° pk(z zo)k-11 (n + p)an(z zo) n+P-1]k=0 n=0

co[E qk(z zo)k-2] [1:±c

k=0an (Z Zo)n+Pi= 0

And after forming the Cauchy products of the series and adding the resultingseries term by term, we get

0.

(zn=0

n+P-2{(n + p) (n + p 1)an + ak [(k + P)Po-k qn-k]1 = 0k=0

(6-35)

Now it follows from the uniqueness property of power series that thisseries will vanish for all values of z in some domain only if the coefficientsof each power of z zo vanish individually. Upon equating these coefficients tozero, we obtain for n=0

and

F (p)ao= 0 (6-36)

n-1F (n + p)an = E ak[(k + P)Po-k

where we have put

176

for n = 1, 2, . . .

(6-37)

F(p) p2 + (Po 1)p + qo (6-3.5)

Since, by hypothesis, ao 0 0, equation (6-36) implies


F(p) = p2 + (p071)p+ qo = 0 (6-39)

This quadratic equation is called the indicial equation. Its two roots arecalled the characteristic exponents of the differential equation at the pointzo. Equation (6-37) is called the recurrence relation.

If the constant p in the assumed solution (6-34) is chosen to be a root ofthe indicial equation (6-38), the coefficient of (z zo)P-2 in equation (6-35)will vanish for arbitrary values of the constant ao. And if there is no positiveinteger n for which F(n + p) = 0, equation (6-37) can be used to calculatesuccessively (starting with ao) the coefficients 78 an in such a way that thecoefficients of all the remaining powers of z zo in equation (6-35) will vanish.Thus, when the constants p and an for n = 1, 2, . . are determined in thismanner, the series (6-34) will at least formally satisfy the differential equation(6-30) with the constant ao arbitrary.

Of course, if F(n + p) vanishes for some positive integer n, we have noassurance that equation (6-37) can be used to calculate an. But let the roots ofthe indicial equation (i.e., the characteristic exponents) be denoted by piand p2 with the notation chosen so that g, pi = p2, and let

Then

v = pi P2 (6-40)

0 (6-41)

And since a quadratic function can always be expressed as the product of itsfactors, we can write

F(p) = (ppt) (1) P2)

Hence,

F(n+pi)=n(n+v) (6-42)

78 Notice that for each integer n the recurrence relation (6-37) expresses ao only in terms of the coefficientsak with 0 k < n. Hence, it can be used to first determine al in terms of ao and then to determine a2 in teems of aland ao. But since al is known in terms of ao, this determines 02 in terms of ao. By proOeding in this manner, each ancall be determined in succession in terms of ao.

177


Then, since equation (6-41) shows that

(n+ v) = n v (6-43)

it follows from equation (6-42) that

F(n+pi) 0 for 1, 2, . . (6-44)

Hence, when p = pi, the coefficients an for n=1, 2, . . . can all be calculatedrecursively in terms of ao from the recurrence relation (6-37). Thus, equatioi.(6-30) will always possess at least one formal solution of the form (6-34) aboutthe point zo.

It is still necessary to verify that this formal solution is justified. In order toCC

do this, we must show that the series E an(z zo)n obtained by the proceduren=0

described actually converges. We can then conclude from the theory of powerseries given in section 5.5 that the formal operations of (1) differentiating theseries (6-34) term by term, (2) forming the Cauchy product of the resultingconvergent series with the convergent series (6-31) and (6-32), and (3) addingthe resulting convergent series term by term are justified 79 and, hence, thatequation (6-34) is indeed a solution to the differential equation (6-30).

In order to establish the convergence of the series in equation (6-34) forthe case where pi, notice that in view of equations (6-42) and (6 -44) therecurrence relation (6-37), with p= pi, can be put in the form

n-iE ak[(k+popn-k+qn_k]k=0an= n(n+ v) for n = 1, 2, . . . (6-45)

79 We shall subsequently encounter (in ch. 9) a case 're the constructed series a'tually diverges and hence., .the formal proccdu

178

Therefore,

1 an 1=


n-IEakkk+ pi)Pn-k+ qn-k]

k=0

In(n+v)I

11 I akl[i Pn-k1 I Pli+ I qn-k1+ kiPn-klik=0

nIn+V.1(6-46)

Let R= min {R1, R2} , where R1 and R2 are the radii of convergence ofthe series (6-31) and (6-32) for p any; q, respectively. Then R is strictly positive(since R1 and R2 are positive) and the Cauchy_ estimates, given in section 5.5,for the derivatives of the analytic functions p and q imply that there exist finitepositive constants M and N such that

MI Pk Rk

which implies that

and I qk I for k= 0, 1, 2, . . . (6-47)

1PkIiP11+1Dclz-51111pil+NRkfor k 1

And since In + vl (n+ v), it follows from equation (6-43) that In+ vl nfor n= 1 , 2, . . .. Inserting these results into equation (6-46) shows that

1 "-I + N kM\ akIan' n E

n R"-kk=0

for n= 1, 2, . . . (6-48)

Now put P = MI pi I+ N + M+ 1. Then P is a finite number which is largerthan 1; and for n 1 and k < n,

1 79


MIP11-1-N-F kM <MIPII+N+Al p


which for n= 1 becomes

Suppose that for n > 1

P lakln E Rn-k

k=0for n= 1,2, . . (6-49)

faller ( laol

ip\klad Ui) laOl

(6-50)

forltc.k<n (6-51)

We shall now show that this implies that the inequality also holds whenk = n. Since n can be any integer larger than 1 and since equation (6-50)shows that the inequality also holds when n= 1, we can then conclude byinduction that the inequality will hold for every positive integer n. Thus,inserting the inequality (6-51) into equation (6-49) shows that

p n-1 Pk

n E laol"k=0(6-52)

But since P - 1, it follows that P"..51)11-1. And when this is inserted intoequation (6-52), we find that

la n1 iaol (6-53)

Hence, we can conclude by induction that this inequality holds for everypositive integer n.

We can now use this inequality to show that the series lb an(z 2.0)711n=

180


which appears in the solution (6-34), converges at least within the circularregion (of nonzero radius)

Iz zoI = fr) (6-54)

To this end, notice that within this circle the inequality (6-53) shows that thenth term of this series has the property that

lanczzoni= lad lzzoin 1 (fOn iaol (Nn = iaoi

And since the series

1laol E

n=0

(6-55)

is simply a geometric series, it certainly converges. Hence, in view of the in-equality (6-55), a simple application of the comparison test " shows that the

02

series E an(zzon is absolutely and uniformly convergent at least withinn=0

the circle (6-54). It, therefore, represents an (single valued) analytic functionwithin this circle.

Thus. we have shown that

wi(z) = (zz.)P. E (in') (Z Zo)nn=0

(6-56)

is a solution to equation (6-30) within some circle about zo for arbitrary atoprovided the coefficients a;;) for n = 1, 2, . . . are computed from the recurrencerelation (6-45). But the series in (6-56) converges within a circle about zowhich passes through the singular point of wi ( earest to zo. In fact, the radiusof convergence of this series must be at least a ge as the distanceR betweenzo and the nearest singular point of equation ( 30). In order to prove thisnotice that R1 and R2, the radii of convergence of the series for p and q, re-spectively, must be equal to the distance between zo and the nearest singular

"See ref. 23.

181


points of these functions. But it follows from the definition of a Sinviar pointof a differential equation that R = min {R1, R2 }. Hence, p and q are singlevalued and analytic within the punctured circular ;$,,Oon A defined by

< Iz zol < R. Now since wi is a solution to the differential equation, itmust be possible (as shown in section 5.9.2) to analytically continue thisfunction along any simple curve in .A. And, therefore, wi cannot have anysingular points within A. But it certainly has a singular point on its circle ofconvergence. Hence, its radius of convergence cannot be smaller than R.And this proves the assertion.

Since the analytic continuation of a solution of the differential equation(6-30) is also a solution of this equation, we can now conclude that the functionwl (z) given by equation (6-56) converges and satisfies the differential equation(6-30) at every point of the domain 0 < Iz zol < R, where R is the distancebetween zo and the nearest singular point of equation (6-30).

We have now shown that equation (6-30) possesses a regular solution(6-56) about any ordinary point or regular singular point of this equation.

&fore obtaining a second regular solution we shall introduce an exampleto illustrate the procedure described in the preceding paragraphs. Thus,consider the equation

3z2ie + zw' (1 + z)w = 0

Since the coefficients p and q are

1 1+ zp(z) 3z q

(6-57)

and since the coefficients (6-28) and (6-29) of the transformed equation are

5z z2(1 + z)2z z2p --= 3- z4q

3

we see that every point is an ordinary point except the point z= 0, which is aregular singular p--)int, and the point z= cc, which is an irregular singular point.

We know that if we seek a solution about the regular singular pointz = 0 of the form

182


w = anzni-pn=0

(6-58)

we will obtain at least one and perhaps two solutions to the differential equationin which the constant ao is arbitrary. And since the nearest singular point ofthe equation to z = 0 is at 00, we know that the series will converge in theentire plane.

Instead of using the general formulas obtained above it is usually easierin any given case to derive the solution from first principles by substitutingthe assumed power series into the differential equation. Thus, after substitutingequation (6-58) into equation (6-57) and differentiating term by term, we get

E[3(n + p) 2 2(k + p) llanzn+P E anzn-FP+1= 0n=0 n=0

In order to collect the coefficients of like powers of z, it is convenient toreindex the sums so that the same powers of z appear in each term. Thus, inthe second sum, we replace the dummy index n by the index k = n + 1 and inthe first sum we put n = k. And since k =1 when n = 0, in the second sum weobtain

[3(k + p)2 2(k + p) llakzk+P ak_izk+P = 0k=0 k=1

Now equating to zero the coefficients of like powers of z gives

and

[3p2 2p l]ao = 0 for k = 0

[3 (k+p)2-2 (k+p) flak= ak_, for k= I, 2, . . . (6-59)

Since ao 0 0, the indicial equation is

F (p)= 3p2 2p I

183


And its roots arepi =1 and p2 = 1/3 (6-60)

When p= 1, the recurrence relation (6-59) can be solved for all values of kand can be written as

alc-1

ak k(3k+ 4)

or writing out the first few terms

for k= 1, 2, . .

an7

ala2= 2 x 10

an n(3n + 4)

In order to determine an in terms of an we multiply together both members ofthese equations to obtain

ai a2 . . an=1

. . .1 2 . . . n) [7 10 . . . (3n +4)] an al an-i

and then divide through by al a2 . . . an-1 to get

anan

n![7 10 . . . (3n + 4) ]for n = 1, 2, . . . (6-61)

Notice that the term in square brackets is a product whose factors areelements of an arithmetic progression. The factors in this progression in-crease by 3 in each term. Hence, if we factor out a 3 from each term we obtain

184


7 77 10 13 . . . (3n + 4) =3n

\ 3+ 1) + 2) . . . (3 + n 1) = 3n (Dn

where we have the generalized factorial function defined in equation (5-33).Upon substituting this into equation (6-61) and then substituting the resulttogether with p = 1 into equation (6-58), we obtain the regular solution

= Ao

where we have put A0 = 3ao.

1 (Z1+1ED° 7 k3/n=0 n1 l3)n

6 2 )

6.3.2 Second Regular Solution

We shall now use the regular solution w1(z) obtained in section 6.3.1 to showthat equation (6-30) always possesses a second regular solution about anyordinary point or regular singular point zo. In order to do this we use the resultsobtained in section 5.9.3, which show that

dz

w2(z)= (z) e

[wi(z))2 (6-63)

is a solution of equation (6-30) which is linearly independent of wi (z). We haveomitted the arbitrary constant c in equation (5-28) since we may assume thatit has been absorbed into the arbitrary constant ao which multiplies w1(z) .

Now upon integrating equation (6-31) along any path in the puncturedcircular region 0 < lz zo I < RI, we obtain

where

P(z)dz= po ln(z zo) + co+ go(z)

go(z) = ifc (zzok

and co is an arbitrary constant.

(6-64).

(6-65)

185


Since the series pk(zzo)k converges, the series (6-65) must also con-

veige (seci;on 5.6). Hence, go(z) is an analytic function in a circle centered atzo. It, therefore, follows from equation (6-56) that

[wi (z)]2

where we have put

Z0)-2p,e-po In (z-zog(z)

= (z ZO)(2P +P°)g(Z)

e-faz)g(z) = e-co and (z) anz zo) n

[gi (z) j2 o=0

(6-66)

Since at" 0 0, there is some neighborhood of the point zo in which g, (z)is never equal to zero And since gi (z) is analytic at zo, it follows that the func-tion [gi (z)]-2 is also analytic and not equal to zero 81 at zo. Similarly, it followsfrom the fact that go(z) is analytic at zo that exp [go(z)] is analytic and non-zero 81 at zo. Hence, the function g(z) is analytic and not equal to zero atzo. It can, therefore, be expanded in the power series

g(z)= cenczzon with ao 0 0 (6-67)n=0

which converges 1n a circle of finite radius about zo.Now since the two roots pi and p2 of the quadratic indicial equation (6-39)

are given by

(Po 1 )-±- V(Po 4go2

it follows that pi + p2 =1 Po. And therefore, in view of definition (6-40) ,

2p1-1-po= v+ 1 (6-68)

81 Notice that [(31(z)]-2 and exp [go(z)] can only equal zero at points where gj(z) and go(z), respectively,become infinite; i.e., at litrigula, points of these functions.

186


When equations (6-67) and (6-68) are substituted in equation (6-66)and the resulting expression is substituted into the integrand of equation(6-63), Ns e find that

W2 (Z) = wl (z)(Z

Zo)v+IE an (z zo) "dz1

n=0

And it follows from the fact that the power series converges that we caninterchange the order of summation and integration (section 5.6) to obtain

w2 (z) =

anw1(z) V (z Zo)nv for p 0 0, 1, 2, .

n=0za n v

z(z) V (z zo)n-v+ tvi (z)a, ln (zzo)n=0n7Ev

(6-69)for v= 0,1, 2, . . .

Since these series converge, they represent analytic functions withintheir circle of convergence. Therefore, the product of either of these serieswith the series in equation (6-56) is again an analytic function which can be

expanded in a power series E a(.2) (z zo) n with a nonzero radius of converg-o=0

ence. Hence, in view of definition (6-40), it follows from equation (6-56) thatequation (6-69) can be put in the form

w2(z) = (z zo)P2 E anz zo) n aw (z) In (zzo) (6-70)n=0

where a= 0 for v 0, 1, 2, . . .

By repeating the argument used in section 6.3.1 to determine the size ofthe circle of convergence of the solution (6- 56), we can again show thatthe function w2(z) given by equation (6-70) converges and satisfies the differ-ential equation (6-30) at every point of the domain 0 < lz zol < R, where Ris the distance between zo and the nearest singular point of equation (6-30).

We have now proved that the linear second-order differential equation(6-30) possesses two linearly independent regular solutions at every ordinarypoint and at every regular singular point.

In order to show how to construct the solution (6-70), it is necessaryto consider certain cases separately.

187

488-942 0 - 73 - 13


6.3.2.1 Case (i): p 0, 1, 2, . . .First, consider the case where thedifference in the characteristic exponents v is not equal to an integer. Thena =0 in equation (6-70), and this solution is of the same form as the trialsolution (6-34) with p= p2. Hence, if equation (6-70) is substituted into thedifferential equation (6-30), we will find, as before (since p2 already satisfiesthe indicial equation), that the coefficient aT must satisfy the recurrence relation (6-37) with p= p2. But since equation (6-40) and the equation followingequation (6-41) show that F (n+ p0= n (n v) , the recurrence relation (6-37)with p= p9 can be written in the form

n-in(n v)a=.5 ak[(k+ p2)pn-k+ qn-k] for n= 1, 2, . . . (6-71)

And since v is not equal to an integer, the coefficient of an never vanishes.Hence, this equation can be solved recursively to determine the coefficientsaq) in exactly the same way that the coefficients ail) in the first solution (6-56)were determined. Nothing new is involved. In fact, in solving the recurrencereation in this case it is usually easier to leave p unspecified as long as possibleand to determine the coefficients of the first and second solutionssimultaneously.

For example, the left member of the recurrence relation (6-59) of thedifferential equation (6-57) considered in section 6.3.1 can be factored toobtain

ak--, for k=1, 2, .) (3p+ 3k+1)

or, writing out the first few terms,

188


aoal= p(3p+4)

a2 (p+1) (3g+7)

an -1an= (p+n-1) (3p+3n+1)

And upon multiplying together both members of these equations we obtain

al a2 a3 an

Hence,

an=

ao al a2 . . .

[p(p+1) . . . (p+ n-1)] [(3p+4) (3p+7) . . . (3p+3n+1)]

ao

[p(p+1) . . . (p + n -1)]n-1)] [(3p+4) (3p+7) . . . (3p+3n+1)]

The characteristic exponents pi and p2 are given by equation (6-60). If 'leset p= pi=1 in this equation, we obtain the coefficients of the first solution.But if we set p= p2, we obtain. the coefficients

189


= ao

[ (1) (+23

3n -4 "6 9 . 3n]. .[3

(÷) ( 1]3

ao[(-1) 2 5 . . . (3n -4)]n!

of the second solution. After using the generalized faclorial function (eq.(5-27)) to get

1(-1) 2 5 . . . (3n- 4)=3'1E1) (-13 ,

F 1)3

(--11-+ 2) (-T +n-1)

=3"(--1)n

and then substituting the results with p= 1/3 into equation (6-58), we findthat the second solution to equation (6-57) is

1 \ n-(1/3)W2 = BO >1: la 1 k 3 )

n=0n

where we have put Bo = bo3-113.6.3.2.2 Case (ii): v= 1, 2, . . . and a = 0. -In this case the second

solution (6-70) will also be of the same form as the trial solution (6-34)with p=p2. Hence, its coefficients 42) for n=1, 2, . . . must again be deter-mined by the recurrence relation (6-71) with p= p2. But since v is a positiveinteger, the left-hand side of this equation will vanish when n= v. The recur-rence relation for a, can therefore not be satisfied unless the right-hand sideof this equation

v=iE ak[(k+popv_k+QP-k]k=0

also vanishes, in which case it will be satisfied automatically for all valuesof a,. We then obtain a solution involving two arbitrary constants. And sincethis solution must be a general solution to the differential equation, it willcontain the solution obtained for p= pi.

190


This situation will always occur in the important special case when zois an ordinary point of the differential equation. For in this case, equations(6-33) and (6-39).show that the indicial equation is

F(p) p(p 1) = 0

Hence, the characteristic exponents are p 1=1 and p2= 0; and their differencev ------ pi P2 = 1 is a positive integer.82 But it follows from equation (6-33)that the recurrence relation (6-71) for al becomes

1 x Oxai=ao[(0Xpi)+0]

and this is automatically satisfied for any choice of al. The recurrence relation(6-71) with n=2, 3, . . . will then uniquely determine the remaining an.We therefore obtain in this case a solution which involves two arbitraryconstants.


(z2 1) w"+ 6zw' + 4w= 0

Since its coefficients p and q are

6zp(z) =z2 -1 q(z) =z24 1

(6-72)

we see that the only singular points of this equation are regular singular pointsat z = 1, z= 1, and = 00.

We know that if we seek a solution about the ordinary point z =0 of theform

w= EanZnn=0

(6-73)

we will obtain a general solution to the differential equation in which thecoefficients ao and al are arbitrary constants. And the series will converge atleast within the circle < 1.

82 That v must be an integer in this case could easily be anticipated from the fait that the characteristic exponentsthemselves must be integers if the solutions are to be analytic at z= zo.

191


Thus, after substituting the expansion (6-73) into equation (6-72), differ-entiating term by term, collecting terms, and factoring the coefficient of z",we obtain

E n(n-1)anzn-2+ (n+4) (n+ 1)anzn=0n=0 n=0

which becomes, upon shifting the index in the second sum,

and

5"x k(k 1)akzk-2+ (k+2)(k-1)ak_2zk-2=0--1=0 k=2

We now equate to zero the coefficients of like powers of z to get

OX (-1)ao=0 1 x0xa1 =0

k(k-1)ak+ (k+2)(k-1)ak-2=r-0 fork=2, 3, . . .

The first two equations merely serve to show that ao and ai are arbitrary, aswe already know. And since the coefficient of ak is not zero for k 2, we canwrite the remaining equations as

ak = k+ 2 ak-2 for k=2, 3, . . (6-74)

This equation shows that each succeeding ak is determined from the a;whose subscript is 2 lower than its own. Thus, this recurrence relation willultimately determine the ak with even values of k in terms of ao and the ak withodd values of k in terms of al. It is, therefore, convenient to consider separatelythe equations for even and odd values of k. First, upon writing out theseequations for even values of k beginning with a2, we find that

192


4cz2 =2

as

6a4 = 4 a2

2(j+ 1)azi a2j-2

Similarly, for odd values of k beginning with a3, we find that

5a3 = 3 al

7a5 =

5

2j +3a2j +1 2j+ a2i-i

Multiplying together both sides of the equations for even values of k anddividing through by a2 a4 as . . . a2;_2 gives

[4 6 .8 . . . 2(j+1)]azi= 2 .4 .6 . . . 2j a°

which becomes, upon removing the common factors of the numerator anddenominator,

193


a2j= (j+1)ao for j= 1, 2, . . (6-75)

And upon proceeding the same way for the odd values of k, we obtain

2 j+ 3a2j+ 1 =

3a1 for j= 1, 2, . . (6-76)

Since equations (6-75) and (6-76) show that the coefficients of even andodd powers of z are given by different expressions, it is convenient to first re-arrange the series (6-73) into two series, one containing the even powers of zand the others containing the odd powers. This is legitimate since everyrearrangement of a convergent powers series converges to the same sum. Thus,equation (6-73) becomes

w= azizzi a2j+ lei+1

3=0 =0

And upon substituting equations (6-75) and (6-76) into this- expression, weobtain the general solution

w= ao E 1)zzi+ E3

2j+3).Z2j+1

to equation (6-72). It is easy to see that both series converge in the circlez I < 1 and diverge outside this circle.

In both the examples given thus far in this section, explicit expressionsfor the general terms in the series were obtained. Also the recurrence relationsin both examples (see eqs. (6-59) and (6-74)) involved only two differentcoefficients; whereas, in general, the recurrence relation will involve n differentcoefficients (see eq. (6-37)). It is, in fact, true in general that there is little hopeof obtaining an explicit expression for the solution unless a two-term recurrencerelation is obtained. This is not necessarily a limitation on the method becauseany recurrence relation can be used to successively calculate numerically asmany terms of the series as desired. In fact, it is in the cases where many-term recurrence relations occur that the general convergence theorems areparticularly useful since, without having an explicit expression for the generalterm, it is not possible to tell from an infinite series itself whether or not it isconvergent.

194


6.3.2.3 Case (iii): v=0, 1, 2, . . . and a0 0. -In this case the Toga=rithmic term will be present. Notice that in case (ii) we did not include v= 0.The reason for this is that when v= 0 the characteristic exponents are equaland therefore there is only a single recurrence relation. Hence, there can onlybe one regular solution of the type (6-34) and the logarithmic term must bepresent in the regular solution (6-70). When v = 1, 2, . . ., a will not be equalto zero if, and only if, the recurrence relation (6-71) cannot be solved for ay. Inthis case, no generality will be lost if we set 83 a 1. The solution can'always bedetermined by substituting equation (6-70) into the differential equation, usingthe fact that wi is a solution to simplify the result, and then setting to zero thecoefficients of the various powers 84 of z zo to calculate the coefficientsa(,;) recursively.

The procedure is best illustrated by considering an example. Thus, thedifferential equation

zw" w = 0 (6-77)

has a regular singular point at z = 0 and an irregular singular point at z= co.We know that this equation will possess at least one solution of the form

te

w = E anzn+pn=0

(6-78)

about the point z = 0 and that this solution will converge in the entire complexplane. Upon substituting the expansion (6-78) into equation (6-77), inter-changing the order of summation and differerdiation, shifting the indices, andcollecting terms, we obtain

(n + p) (n + p -1)anz"+P-1 (n + p)an-izn+P-1= 0n=i.-111 n=1

Equating the coefficients of like powers of z to zero gives for n = 0, since aois arbitrary,

p(p 1) =0

Siace wi is detemined only to within a constant factor.8.1 The logarithmic terms will always cancel out.

(6-79)

195


and

(n p 1)an = an -1 for n = 1, 2, . . . (6-80)

The roots of the indicial equation are p1= 1 and p2 = 0; hence, the differenceof the roots v = pi /12 = 1 is an integer. First, consider the case wherep = pi= 1. The recurrence relation is (see eq. (6-76))

an = 1an -1 for n = 1, 2, . . . (6-81)

After writing out the various terms of equation (6-81), multiplying the cor-responding members of these terms together, and dividing out common factors,we get

1 1an = ao = ao

1 2 3 . . . n n.

Hence, we find that the solution for p=1 is

VIc° 1 n+1W1 al) 2a Zn=0

which can easily be summed to obtain

W aozezIT

Next, consider the case where p = p2 = 0. The recurrence relation is(n 1)a = an-i. Since ao 0 0, it is clear that this equation cannot be solvedfor a1. Hence, the differential equation must possess a logarithmic solution.We, therefore, seek a second solution of the form

(6-82)

(6-83)

CO

W2 = E anzn + w1 (z) In zn=0

(6-84)

Upon substituting this expansion into equation (6-77) and recalling that wisatisfies the differential equation, we find that the resulting coefficient of In z

196


vanishes. Then shifting the index and collecting terms in the summation gives

( 2w; --1z /4)1 [n(n-1)a" nan-dzn-i= 0n=i

Hence, after substituting in equation (6-83), we get

an(l+z)ez+ E [n(n-1)annan_I]z"-'=0n=1

(6-85)

(6-86)

ceBut replacing ez by its series representation V 1-zn and then shifting theindices shows that n=0 n!

,esao zn-1+ E [n(n-1)a nan_dz"-1=0E1=2 (n-2)!zn-i+ E (n -1)!°1)!En=1

Then upon equating to zero the coefficients of like powers of z, we find that

and

ao :1= ao for n=1

aoan n-1 (n-1)(n-1)!

for n=2, 3, 4, . (6-87)

where ao and at are arbitrary. Actually, we could simplify matters by settingal= 0, but we shall carry at through as an arbitrary constant in order todemonstrate this.

Upon writing out the first few terms of equation (6-87), we get

aoa2

1 1 1 !

az ao ao (1 1\az

2 2 2! 1 2 if VL 2)

197


a3 ao a1 ao (1 + 1 1)

a43 3 3! 1 2 3 -3! 2

+3

Hence, we conclude by induction that

a1aan ° Hn-1 for n = 2, 3, . . (6-88)

(n 1) ! (n 1) !

where we have defined Hn to be the partial sum of the harmonic series. That is,

0 for n = 0Hn =7-

1 n1+2+-

3+...+n = k

kk=1

for n -= 1, 2, , .

Substituting equations (6-83) and (6-88) into equation (6-84) yields

= ao [(zez In z) + 1 _1 Hn_izd1+ ai 2, 1)! z

n=2 n=1

which becomes, after shifting the indices in the sums,

wz = ao [(zez In z) + 1 E 1Hnzn+1] + ai

n=1

Izn+i

zan=0

n1

(6-89)

Notice that the second sum is essentially the solution w1. Hence, the term insquare brackets must be a solution which is linearly independent of w1. This(i.e., the bracketed term) is the solution we would have obtained if we put a1= 0.

6.3.3 Summary

We have now shown how two linearly independent solutions to the differ-ential equation (6-30) with analytic coefficients can be found in the neighbor-hood of any regular singular point zo. First the indicial equation (6-39) is

198


determined. If the difference v of the two roots of this equation is not aninteger (including zero), two solutions of the form (6-34) are obtained, onefor each root of the indicial equation. However, if v is an integer, the recurrencerelation (6-37) for the term ap must be investigated with p= p2. It will eitherautomatically be satisfied for any a,,, or it will be impossible to satisfy for anyay. If it is automatically satisfied, ao and ai, will be arbitrary; and we willobtain two linearly independent solutions with p= p2. If v= 0 (i.e., pi= p2)or if i, is an integer and it is impossible to satisfy the recurrence relation, thesolution will have the form (6-70) and can be found by substituting this forminto the differential equation. The point at infinity is treated by making thechange in variable z=

6.3.4 Computation of Indicial Equation

Since the nature of the solutions of equation (6-30) in the neighborhoodof a regular singular point zo is so dependent on the roots of the indicial equa-tion (characteristic exponents), it is useful to be able to determine this equa-tion without first finding the expansions (6-31) and (6-32) of the coefficientsabout zo.

Equation (6-39) shows that this is accomplished once po and qo are known.But it follows from equations (6-31) and (6-32) that

Hence,

(z zo)p(z)= po+ pk(zZO)kk =1

zo)2q(z)= q0+ E qk(zZO)kk=1

po=lim [(z zo)P(z)] and q0= lim [(z zo)2q(z)]

Thus, for example, the equation z2wu+z(2 + 3z) w' + (1 z)w= 0 has a regularsingular point at z= 0 and

P(z) =--2 +3z and q(z) = 1 z

Z2


Hence,

2 3zpo= lim z (--) =2 and go= lim z2

z-0 Z. z-0

and the indicial equation (6-39) is

p2+(po-1)p-1-q0=p2+p-F1=0

If the regular singular point zo is the point at infinity, we make the change invariable'z=1/4 and investigate the point 4=0. This leads to the relations

Po=lci_mo {4 _ j_-t2 et)1}

q°' li-T) 42 [t4 q (Di

And upon changing back to the ori,;inal variable z, they become

Po= lim [2 zp (z)]

200

and qo = lirn z2q (z)

CHAPTER 7

Rieman11-Papperitz Equation and the

Hypergeometric Equation

7.1 THE FUCHSIAN EQUATION

Having shown how to construct solutions to a differential equation withanalytic coefficients about its regular singular points, it is natural to studythose equations whose only singularities (in the extended plane) are regularsingular points. Such equations are called F uchsian equations. We shall restrictour attention to Fuchsian equations of the second order; that is, equations ofthe form

w" p(z)u1 + q(z)tv=-- 0 (7-1)

This equation can have, at most, a finite number of singular points. In order toprove this, notice that since the only singularities of the coefficients p and q inthe extended plane are poles, these functions must be rational (see section5.4). But since the number of singularities of a rational function which occurat finite points of the plane is equal to the number of distinct zeros of its poly-nomial denominator, it follows that p and q have, at most, a finite number ofsingular points. However, every second-order equation must have at least onesingular point. For if equation (7-1) had no singularities in the extended plane,the coefficients p and q would be everywhere analytic; and, hence, by Lion-vine's theorem, they would be constants. However, even if the constant value ofp were zero, the coefficient 2z z2p(z) of the transformed equation (seeeq. (6-28)) would still have a simple pole at infinity. But this would contradictthe assumption that equation (7-1) had no singularities. Hence, we concludethat this equation must have at least one singularity.

201


Since the rational function p has, at most, simple poles and the rationalfunction q has, at most, poles of order 2, it is clear that any Fuchsian equationwith not more than m singular points in the finite plane must have coefficientsof the form

p (z) =P(z)

(z z (z z2) . . .

Q (z)q(z) =

(z z 1)2 (z z2)2 .

where P and Q are polynomials.

7.1.1 Fuchsian Equations With, at Most, Two Singular Points

There are two possible forms which a Fuchsian equation with, at most,two singular points can have. First, consider the equation which has, at most,one singular point in the finite plane (with a possible singular point at infinity).Its coefficients must take the form

P (z) Q (z)P (z) a

q(z)(z a) 2

But if the coefficients of the transformed equation 2z z2p and z4q are to haveat most a simple pole and, at most, a pole of the order of 2, at infinity thepolynomials P and Q must both be constants, say A and B, respectively. Hence,the differential equation must be of the form

wit + Aw' + w 0z-a (z a)2 (7-2)

Next, consider the case where the singular points are both in the finiteplane. Then the coefficients must be of the form

P(z) Q (z)(z a) (z b) (z a)2 (z b)2

But the coefficients 2z z2p and z4 q of the transformed equation will be analytic

202

RIEMANN-PAPPERITZ AND HYPERGEOMETRIC EQUATIONS

at z= 00 only if there are constants C and D such that P = 2z + C and Q = D.Hence, in this case the differential equation must be of the form

2z + C+(z a) (z b)

+(z a)2 (z b)2 (7-3)

Thus, a Fuchsian equation which has, at most, two singular points must beeither of the form (7-2) or of the form (7-3). It is easy to verify that thegeneral solution of equation (7-2) is

(z a)P1 + C2 (Z a)P2 if (A 1)2. 4B 0w=

(z a)Pi [CI + C2 111(Z a) ] if (A 1)2 4B = 0

where CI and C2 are arbitrary constants and pi and p2 are the roots of theindicial equation p2 + (A 1)p+B=0. Since the change in independentvariable t =1/ (z b) transforms equation (7-3) into an equation of the form(7-2), it follows that the general solution to equation (7-3) can also be ex-pressed in terms of elementary functions.

Thus, the solutions of any Fuchsian equation which has no more thantwo singular points in the extended plane are elementary functions.

7.1.2 Fuchsian Equations With, at Most, Three Singular Points

In order to obtain a Fuchsian equation whose solution is not elementary,we must consider an equation which can have three regular singular points.If we require first that these three points, say a, b, and c, all lie in the finiteplane, the coefficients of the differential equation must have the form

P(z)P (z)

(z a) (z b) (z c)q(z) = (z)(z(z 02 b)2 (z c)2

where P and Q are polynomials. And since the point z = cc is to be an ordinarypoint, the coefficients of the transformed equation

2zz2P

(z a) (z b) (z c)

488-942 0 - 73 - 14

and z4Q(z a)2 (z b)2 (z c)2

203


must be analytic at z= co. But this can occur only if P and Q are quadratic inz and the coefficient of z2 in P is 2. Since the degree of the numerator is lessthan the degree of the denominator, it follows from the elementary theoryof partial fractions that there exist constants Aa, Ab, Ae,Ba,Bb, and& such that

and

P(z) Aa Ab Ac(z a) (z b) (z c)= za+zb+zc

Q(z) Ba Bb _L. Be(z a) (z b) (z c) z a zb zc

The differential equation, therefore, takes the form

d2w (A A )dwa Ab c

de za zb zc dzBa Bb (7-4)amzb z (z a)(z b)(zc)

And since the coefficient of z2 in P(z) must equal 2, it follows that

Aa+ Ab+ 'lc= 2 (7-5)

Otherwise the constants Aa, Ab, Ae, Ba, Bb, and Bc are arbitrary.Now at each of the points a, b, and c there is a pair of characteristic

exponents, say (a', a"), (p' , (3 "), and (y', y") ,85 respectively. In order to finda relationship between these exponents and the A's and B's, notice that since

Po = lim (z a)p(z) = lim [Aa + Ab (l + Ac A a_z b z c

'4.5 The pairs (a', a"), , a"), and (y', y") are solutions of the indicial equations at the points a, b, andc, respectively.

204


and

qo = llin (z a)2q (z) = 1im[B + Bb (Z +(z al (z1

sez Z b z cJJ (z b) (z c)

B a

(a b) (a c)

at z = a, the indicial equation at this point is

Bp2± 1)p + = 0(a b) (a c)

But it follows from the properties of the roots of quadratic equations that theroots a' and a" of this equation satisfy the relations

a' + a" = 1 Aa a, allB

(a b) (a c)

And we can show in exactly the same way that

bf3' + g" = 1 A bPi 13" (b aB)(b c)

y' + y"= 1 A,BcYY (c a)(c b)

Upon using these relations to eliminate the A's and B's in equation (7-4),we arrive at the Riemann-Papperitz equation

d2w (1 a' a" +1 is" 1 y' y"\ dw +I a' a" (a b) (a c)dz2 z a z b z c / dz L z a

+', (3" (b a) (b c) +ycy" (c a) (c b)1(z a) (z w b) (z c)_z b z c j

(7-6)

205


Eliminating the A's in equation (7-5) shows that the characteristic ex-ponents at the singular points of equation (7-6) satisfy the relation

al at? pt ,),?? 1 (7-7)

More generally, the sum of all the characteristic exponents at the singularpoints of any second-order Fuchsian equation is two less than the number ofsuch points. This relation is called the Fuchsian invariant for the equation oforder 2.

In a similar way, it can be shown that if a Fuchsian equation with, at most,three regular singular points, has one of these points, say c, located at z =00,this equation must be of the form

d2w+ (1 a' 1I3'g"\dwdz2 z a z -b

/3

dz

±r a' a" (a b) + 1313" (b a) , wL z a zb ± 7

1

(z aj ) (z--b)=0 (7-8)

where y' and y" still denote the characteristic exponents at c and equation(7-7) still holds. Since equation (7-8) can be obtained by formally takingthe limit c> co in equation (7-6), we can say that equation (7-6) holds evenwhen c= 00. Then with this understanding, equation (7-6) is the most generalFuchsian equation which has, at most, three singular points.

7.2 RIEMANN P-SYMBOL

When w is a solution of the Riemann-Papperitz equation whose singulari-ties occur only at the distinct points a, b, and c, we shall sometimes write

b c

/3" 7"

(7-9)

The right side of equation (7-9) is called the Riemann P-symbol. The char-

206


acteristic exponents of the differential equatlisn are listed below the correspond-ing singular point. (Any of the singular points can be at itzfinity.) These ex-ponents must, of course, satisfy the Fuchsian invariant

a' + a"+[3' +13"+ y' y"= 1

Finally, the independent variable is located in the fourth column.If we interchange any of the first three columns in the P-symbol (7-9),

the notation still refers to the same equation, There are 3!= 6 ways in whichthis can be done. Similarly, interchanging the order of the exponents in anygiven column leaves the meaning of the symbol unchanged. Thus, for eacharrangement of the points a, b, and c, there are eight arrangements of theexponents. Hence, there are 6 X 8 = 48 different P-symbols which refer to thesame equation and, therefore, have the same meaning.

We shall now show that

a

a

a' 13' z =-- e licP i) ' k 13'

al, a" k P"

when a, b, and c are finite and

b 00

)z = (z a)ka9 a' k /3' y' +k

' k 3" y"-I- k

b

y'+ky"+ k

00

(7 10)

(7 11)

when c =Equation (7-10) means that if w is a solution of a Riemann-Papperitz

equation and if we make'the change of dependent variable

a kW- UZ-C (7-12)

207


in this equation, then the transformed equation (which has u as the dependentvariable) is also a Riemann-Papperitz equation and the location of the singularpoints is left unchanged by this transformation. However, the ,Liaracteristicexponents at the points a and c are altered in the manner indicated by thesymbols. Equation (7-11) is the corresponding transformation for the casewhere one of the singular points is at infinity.

In order to prove that equation (7-10) holds, we first substitute equation(7-12) into equation (7-6). Then upon noting that

,(za)k[u

, k (ac)w

z c (z a) (z c)

(zlk [le 1 1 , k (a c) (lc 1 k +1w ZIC 1-

Z C a zc (z a) (z c) a zc) j

and

1 1 ((za) (z c) (z X) (za) (z b) (z c)

1-1-z X

for X= a, b, or c

we find after combining terms that

d2ui 11 a' a"+2k± 1 --(3'--pu+1-y'-" 21c\ dudz2 ,k za z b z ) dz

[(a' k) (a" k) (a b) (a c) /3'f3" (b a) (b c)

z a z b

I+k "+ k ( -a

z C (z a) (z b) (z c)

But comparing this with equations (7-6) and (7-9) shows that equation(7-10) holds. Equation (7-11) follows from equation (7-8) in the same way.

208


Having considered the transformations of the Riemann-Papperitz equa-tions under a change of dependent variable, we shall now show how this equa-tion transforms under a change of independent variable. In this case, thechange of variable which transforms a Riemann-Papperitz equation into anotherRiemann-Papperitz equation is the nonsingular linear fractional transformation

Az+Bt---= Cz+D whereAD- BC 00 (7-13)

studied in chapter 5.It is again convenient to describe this transformation in terms of the Rie-

mann P-symbol. Thus, if t is the change of variable (7-13), then

P a' f31 y' z =P a' 131 y'

,y1 an

(7-14)

where al, b1, and el are the images of the points a, b, and c, respectively, underequation (7-13). For example,

Aa+BCa+ D

In order to prove equation (7-14) we first recall that (see section 5.3) per-forming the transformation (7-13) is equivalent to performing in successionnot more than four elementary transformations, each of which has one of theforms

t = iz (7-15)

t= k+ z (7-16)

1t=-z (7-17)

It can now be verified, by substituting in turn each of equations (7-15) to (7-17)

209


into equation (7-6), that in each case the transformed equation is a Riemann-Papperitz equation with the same characteristic exponents as equation (7-6)and that the singular points of the transformed equation are the images underthe transformation of the original singular points. Hence, equation (7-14) holdsfor each of the elementary transformations (7-15) to (7-17). But since thegeneral nonsingular linear fractional transformation is equivalent to performinga succession of these transformations, it is clear that equation (7-12) must holdfor this transformation also.

7.3 TRANSFORMATION OF RIEMANN-PAPPERITZ EQUATION INTO HYPER-GEOMETRIC EQUATION

We shall now show that the general Riemann-Papperitz equation (7-6)can always be transformed by the application of a number of transformations(each of which has one of the forms (7-10), (7-11), and (7-14)) into a certainstandard equation (canonical form). The usefulness of this result is principallydue to the fact that the transformed equation, which has a regular singularpoint at the origin, has a two-term recurrence relation at this point. As we haveseen, this allows us to find an explicit expression for the series solution.

In order to accomplish this reduction let w be any solution to equation(7-6). Then

(aP cr

al

b c

13' Yi

where we suppose that the notation has been so arranged that a and b arealways finite. We first introduce the new dependent variable v by

lzary

if c 0 00(z cw=

(z a)(1' v if c =00

and then use equation (7-10) or (7-11) to show that

210

(7-18)


a

v:= P 0 13" y'+a'(a__ y+a'

We again change the dependent variable by using the transformation

b\

V

(z b)13' u

if c 00

if c= co

(7-19)

(7-20)

and then apply equation (7-10) or (7-11) to equation (7-19) to show that

a

u= P 0

a'

0 y' +a' -1-13'

pl yll at

(7-21)

Now let us use the linear fractional transformation (discussed in section5.3) which maps the points a, b, and c into the points 0,1, and co, respectively,to change the independent variable. Thus, a new independent variable t isintroduced by

t=iz\ (bc\t--(b a) (z c) (7-22)

(When c= 00, we understand t to be the limit of this expression as c-0 cc,which is simply the first term in the right-hand member.) And in view of equa-tion (7-14) it follows from equations (7-21) and (7-22) that

ll

0 1 00

0 0 y' + a' + p'

II Ia f3"f3' 11+a' -FP'

(7-23)

211


The transformations (7-18), (7-20), and (7-22) can now be collected intothe single formula

a' (zbyru(t)

t=1Z a\ (\zz c f \z c) t b a) (zc)w(z)=

z(za)a. (zb)19' u(t) t=b

aa for c= co

for c 0 oo

(7-24)

The function u given in equation (7-23) is the solution to a Riemann-Papperitzequation which has only four nonzero exponents. Since the sum of these ex-ponents is unity, they can be characterized by three constants, say a, [3, andy. It is customary to define these constants by

a=a'+131+), ±pl

Then equation (7-23) can be written as

u=P(0 1 co

0 0 a1y yap

y= 1 + a' a"(7-25)

(7-26)

But comparing equations (7-8), (7-9), and (7-26) shows that the Fuchsianequation satisfied by (7-2(5) is

d2u (1 y) 1 (y 0 +a1 udt2 t1 dt pt t 1 itti=n1.

And upon rearranging this equation slightly, we obtain the hypergeometricequation of Gauss,

u dut (1 t) d2 + [y (a+P-1-1) dt apt= 0 (7-27)

Its solutions, which will be obtained subsequently, have been extensivelystudied.

212


We have, therefore, shown that any Riemann-Papperitz equation can betransformed into the hypergeometric equation (7-27) by introducing a changeof variable of the type (7-24).

Hence, once the solutions to the hypergeometric equation (7-27) havebeen found, we can find the solutions w to any Riemann-Papperitz equationsimply by using the transformation (7-24) to express them in terms of the solu-tions to the hypergeometric equation.

However, in any specific case, it is usually more convenient to rederivethis transformation by using the Riemann P-symbol. Thus, consider theRiemann-Papperitz equation

For this equation,

2z2(z-2)w"-z(3z-2)w' +2(z -1)w= 0 (7-28)

p(z) = 3z-22z(z -2)

z -1q(z) z2(z -2)

Hence, its singularities are located at the points z = 0, z--= 2, and z= 00.At z =O

3z 2 1_ J.po= lim

z--,o{ 2(z-2) j 2go= limz- 1 =-1.-.0 z -2 2

and, therefore, the indicial equation at this point is (see eq. (6-39))

The roots of this equation are 1 and 112.At z--=2

3z-2 (z 2) (z -1)po= lim 1

z-.2 LZ z.--*2 Z2

and, therefore, the indicial equation is

p2 -2p =0

213


The roots of this equation are 2 and 0.At z=00

Po= lim [2 zp(z)]=---- li [ 2+ 3z ] =7-2(z-22 ) 2

go= lim z2q(z)= limz 1-- 1

and, therefore, the indicial equation is

p2+-2p+1=0

The roots of this equation are 1/2 and 2. Hence, the solutions to equation(7-38) can be denoted by

2

/2 0 2It now follows from equation (7-11) that

0 2 00

w= z 1I2P ( 0 0 0

1/2 2 3/2

And it can be concluded from equation (7-14) that

0 1 00

w=-- z'I2P 0 0 0

1/2 2 3/2

214


where t = z/2. But upon comparing this with equation (7-26), we see that

a =0 3=

1= (7-29)

Thus, the solutions to equation (7-28) can be expressed in terms of the solu-tions to the hypergeometric equation (7-27) with the constants a, /3, and ygiven by equation (7-29).

7.4 HYPERGEOMETRIC FUNCTIONS

It remains to determine the solutions to the hypergeometric equation(7-27). We shall first seek a solution about the regular singular point z =0.Since the P-symbol for equation (7-27) is given by equation (7-26), it followsthat the characteristic exponents at z= 0 are 0 and 1 -y. Suppose first that yis neither zero nor a negative integer. Then the methods developed in chapter 6can be used to obtain a power series solution 86 of the form

u= E a ntnn =0

(7-30)

which corresponds to the characteristic exponent zero. Upon substituting thisinto equation (7-27), collecting terms with like powers of t, and shifting theindex n, the lowest one present, we obtain

n(y+n-1)ant"-1 (a+n-1)(g+n-1)an_itn-1 = 0n = 0 n = 1

Hence, the n=0 term shows that ao is arbitrary and we obtain the two-term recurrence relation

(a+n-1)(P+n-1)an= an -1 forn=1,2, . . (7-31)n(y+n-1)

where the division by n(y+ n-1) is permitted since we have required that ybe neither zero nor a negative integer and, therefore, the denominator can nevervanish. Upon writing out equation (7-31) for successive values of n, multiply-ing the corresponding members of these equations together, and dividing

86 Recall that it was shown in section 6.3-that the power series solution corresponding to the largest characteristicexponent always exists.

215


out common factors, we find that

an=[a(a+1) . . . (a+n-1)][13(13+1), . . . (p+n-1)]

n![y(y+1) . . . (y n 1)]

And by using the generalized factorial function defined in equation (5-33),this can be written in the more concise form

(a) n 03) n1an

71-lvY)n a°

Hence, the solution (7-30) is

u= aoF(a, 13; y; t) (7-32)

where we have put

F (a, p; y; t) (u)n(i3ln0

n!(y)n for y 0 0, 1, , . . . (7-33)

The function F(a, f3; y; t) is called the hypergeometric function, and thenotation used herein is universally accepted. The reason for calling thisfunction the hypergeometric function is that when a =1 and p=y, this series(7-33) reduces to the geometric series. Thus,

1F(1, y; y; t) =1tIn fact, it follows from the binomial theorem that

F(a, y; y; t)=F(Y, a; y; t)= (1_10a

for any number a.

216

(7-34)


Notice that if either a or 13 is zero or a negative integer, the series (7-33)terminates after a finite number of terms and, therefore, represents an entirefunction. In all other cases, a simple application of the ratio test (ref. 23) willsuffice to show that it converges in the circle It1 < 1 and diverges outside thiscircle.

We have, therefore, shown that, provided y is neither zero nor a negativeinteger,

u=F(a, y; t) (7-35)

is a solution to the hypergeometric equation

t(1t)u"--F [y (a- F p-Fot]u' -a1u.0 (7-36)

at least in the circle It( < 1.Since equation (7-36) is a Riemann-Papperitz equation, its solutions u

can be denoted by the P-symbol

0 1 co

u = P 0 0 a

y y a 13

(7-37)

Also, inasmuch as the order in which the exponents are listed in a given columnis immaterial, we can interchange those in the first column to obtain

0 1 00

u = ]. y 0

0 y a 13


a

217


(

0 1 00

u= t!-TP 0 0 a + 1- y

y-1 y-a-I3 P+1-y

(7-38)

Notice that the P-symbol in equation (7-38) is in canonical form. It, there-fore, denotes a solution to the hypergeometric equation whose exponents atthe points 0,1, and 00 are 0, y 0, y a -0; and a + 1- y,13 + 1- y, respec-tively. And we can choose this solution to be the hypergeometric function F.Hence, upon comparing equations (7-35) and (7-37) with the P-symbol in equa-tion (7-38), we find that

u= tl-YF(a+ 1 -y, /3+1 -y; 2 -y; t) (7-39)

is also a solution to equation (7-36) provided, of course, 2 -1/ is neither zeronor a negative integer. Since the hypergeometric series converges for Id < 1,the function (7-39) must satisfy the differential equation in this region. Noticethat if y= 1, the solutions (7-35) and (7-39) are identical. In fact, equation(7-37) shows that, in this case, equation (7-36) has equal exponents at t =0and, therefore, one of the solutions composing the canonical basis at t = 0 mustcontain logarithmic terms. Next, if y is any integer other than unity, eithery or 2 y is either zero or a negative integer. Thus, either the hypergeometricfunction (7-35) or the hypergeometric function (7-39) is undefined. The remain-ing equation provides a solution to the differential equation. Finally, if y is notan integer, both equations (7-35) and (7-39) are solutions to the differentialequation in the circle It' <1. Since the series expansion for equation (7-35)begins with t° and that for equation (7-39) with t'-7, they must also be linearlyindependent solutions.

Interchanging the exponents in the second column of equation (7-37) leadsin the manner described above to the solution

u= (tF(y-- p, y-a; y; t) (7-40)

in the domain J tj <1. And interchanging both the exponents in the first columnand those in the second column leads to a fourth solution

u= (t -a-0F (1_p, 1-a; 2-y; t) (7-41)

218


in the domain It' < 1. But, at most, two of these four solutions can be linearlyindependent. If y is not an integer, there exists a linear relation connectingany three of them. And, if y is an integer, the two solutions which exist can differonly by a multiplicative constant.

Another set of four solutions can be obtained by interchanging the points0 and 1 by means of the transformation z= 1 t. Thus, for example, equation(7-37) is equivalent to the equation

(1 0

u= P y-a-13 1-y

0 0

a t

)Upon applying equation (7-11) to this equation twice in succession, we obtain

(c-F-fl-y y-1 1-a

1 CO

u= 0 1 -/3

It now follows from equations (7-13) and (7-14) that the transformation z= 1 -tmaps the points 1, 0, and co into the points 0, 1, and co, respectively, and leadsto the expression

1 co

u= (t--0)---at1-)P 0 0 1-13 1 tal py y-1 1-a

Since this P-symbol is in canonical form, it represents a solution to ahypergeometric equation which we can again choose to be F. Thus, we findafter comparing exponents with equations (7-35) and (7-37) that

u= (t - W.-a-se-T(1 -13, 1 a; 1 + y- a -P; 1 t) (7-42)

219

488- 942 0 - 73 - 15


Notice that the hypergeometric series in the solution (7-35) converges inthe circle jtj < 1, while that in the solution (7-42) converges in the circleIt 11 < 1. By systematically exploiting the technique used above, Kummer(1810 to 1893) obtained 24 solutions to the hypergeometric equation (7-36),each of which is expressed in terms of the hypergeometric function F. Ineach of these solutions the variable is one of the quantities t, 1 t, (t 1)/t,t/ (1 t) , 1/t, and (1 0-1. Since the function F with variable t convergesin the unit circle Iti < 1, these solutions will converge in one of the regionsshown in figure 7-1. Each series will converge in the appropriate shaded regionand diverge outside it unless the series teminates; in which case, it willconverge in the entire plane. Of the 24 solutions, four converge in each ofthe regions shown. Of course, only two solutions can be linearly independentin any given region. Some of the regions shown in the figure overlap. Hence,there are numerous relations among the various 24 solutions obtained byKummer. These relations not only provide useful identities among hyper-geometric functions but also provide formulas which can be used to ana-lytically continue the solutions of the Riemann-Papperitz equation from oneregion of the plane to another. Thus, it can be shown, for example, that wheny is neither zero nor a negative integer, the identity

F (a, P; y; t) = (1 t)Y-a-oF (y a, y f3; y; t) (7-43)

holds in the domain Iti < 1. An example of an identity which holds in the inter-section of the regions shown in figures 7-1(a) and (b) is

f for Itj < 1 and It < 11F (a, 3 (

It is frequently convenient to express the hypergeometric function as anintegral. This can be done when A y>Ap>0 and Iti < 1. In order to arriveat this result, we substitute equation (5-35) into equation (7-33) to obtain

r (y) (a)» r (13+ n)F (a, P; Y; t) r (13) (y + n) t

220

(7-44)


9-t

(a)

Ic)

(el

(a) The region 0 --5 111 <1.

pat

pat

(c) The region 0 11 <1.

(b)

()

(b) The region 0 < 1-1 I < 1.

pat

At

pat

(d) The region 0 < 1-1

t< 1.

(e) The region 0 < 1. (f) The region 0 r--11 < 1.

FIGURE 7-1.Regions of convergence for hypergeometric series.

221


Since it follows from equations (5-37) and (5-38) that

(/3 +n) r (y- P) - (p+ n, y- P).= 743+"(1-- T)Y -a-1dTr (7+ n)

forAy>..4p>0

equation (7-44) can be written as

F (a, 13; y; t) =(a)n 11

r(a) r -p)i.,0 r17 t" To+n-1(1 - T)r-ft-1dr

Upon interchanging the order of integration and summation (see section5.6), we find that

(y)743-1 - TYY-13-1F (a, Y; t)

r(y)r (y °

(a),, ,

iE ) n aTn=0 nt

But the series can be summed by using equations (7-33) and (7-34) to obtainthe integral representation

F Fey)TO- dT(a, P; Y; t) r(f3) r (y i3) j0

for Ay >ge.p>Oand Iti <1

which was discovered by Euler. Since this integral exists and represents ananalytic function for all values of t except where t is real and larger than 1,it provides an analytic continuation of F(a, /3; y; t) from the unit circleItI < 1 to the entire t-plane cut along the positive real axis. Barnes obtainedthe more general integral representation

1 r(y) r (a+ z) r (p+z) r ( z)(F (a, (3; y; t) 27ri r(a) r(p) ,_1i. r (y+z ) dz

where larg (-01<lr and the integration is to be carried out along any pathwhich lies to the right of the poles of r(a+z) f(p+z) and to the left of thepoles of r(z).

222


The six functions F(a± 143; y; t) , F(a, #± 1, y; t), and F(a, f3; t)are said to be contiguous to F(a, /3; y; t). Gauss proved that there is a linearrelation between F(a, /3; y; t) and any two of its contiguous functions, thecoefficients being linear polynomials in t. Some examples of these relations are

y(1-0F(a,/3; y; t) = (13 y)tF (a, p; y +1; t) yF (a 1, 13; y; t)

0-0F(a,13;y; t)= PF (a, 13+1; T; t)-4F(a+1, /3; t)

The derivative of a hypergeometric function is also a hypergeometricfunction. In order to show this, notice that the series (7-33) can be differentiat-ed term by term within its circle of convergence to obtain

dF (a)(p)ntn-1dt= (y).(n-1)!

Upon shifting the index by putting k= n-1, this becomes

dF (a)k+i (13)k+1dt= (y)k+ik! t 7-45 )

But since it follows from definition (5-33) that (a)k+i = a(a + 1)k, (p)k+.=/3(/3 +1)k, and (y)k +1= y(y +l)k, equation (7-45) shows that

dF («, ; t)F(a+1,13+1;y+1;t)d

More generally, the Jacobi identity

(7-46)

d"[1.4- /3; y; t)] = (cx)nta-lF (a + n, /3; y; t) 7-47)

dtn

can be established by multiplying both sides of the identity by t'-a and thenverifying that the left side satisfies the hypergeometric equation with param-eters a+ n, p, 7.

223


There are many other relations connecting the hypergeometric functions.The reader can find these tabulated in reference 26 and discussed in somedetail in reference 27. Many of the known functions, such as the elliptic inte-grals of the first and second kind, the inverse sine, etc., can be expressed ashypergeometric functions.

7.5 GENERAUZED HYPERGEOMETRIC FUNCTIONS

The concept of hypergeometric function can, itself, be generalized. Tothis end, notice that in the definition of the hypergeometric function

F (a, (3; y; z) = (").(P)."=o (7)nn!

the numerator and denominator of the coefficient of z " /n! contain only productsof generalized factorial functions. Hence, it is natural to define the generalizedhypergeo metric function p Fq by

PF Q(al as, al3; ') y2,\ k--xN (01)n(a2)n 004"

*9 'YQ; z 2, n! (yi)n (Y2)n (7q).n=0

(7-48)

provided the series converges." The numbers p and q are nonnegative integerswhich denote the number of parameters in the numerator and in the denomi-nator, respectively. Thus, the ordinary hypergeometric function F is a 2F1Since the parameter /3 cancels out in the numerator and deno-'tor inequation (7-34) , we see that

1

(1-t) (a; ;t)°'

where we indicate that there is no parameter in the denominator byinserting a dash. The function 1F1 is treated in the next chapter. The generalizedhypergeometric functions are studied in considerable detail in reference 27.

"This series will always converge in the unit circle if q= p+ 1 and it will converge in the entire plane if q > p+ 1.The other cases are more complex.

224


7.6 LOGARITHMIC SOLUTION TO HYPERGEOMETRIC EQUATION

We have found two linearly independent solutions to the hypergeometricequation (7-36) in a neighborhood of t= 0 for the case where y is not aninteger. However, when y is an integer, only one solution has been found.The remaining solution can be found by the methods of chapter 6. Thus, forexample, when y is a positive integer, u1= F (a, (3; y; t) provides one solutionto the hypergeometric equation (7-36); and the methods of chapter 6 showthat if neither a nor 13 is a positive integer smaller than y, a second linearlyindependent solution is given by

u2= F (a, p; y; 1) ln tn! (1y),i

n=0 (1 C)n+I(1 P)n+1t-(n+1)

(a)"(13)" H a n) +H (P,n)--11 (y,,n)H (1,n)] t",n=1 (y )n

where the finite sum is to be omitted when y=1 and we have put"

n-111(a. n)

k=0 " n

We can obtain the second solution for the case where y is zero or a negativeinteger in the same way by starting with the solution

--= ti-YF (a+ 1 y, 1 y; 2 y; t)

7.7 SPECIAL RIEMANNPAP?ERITZ EQUATIONS

The classical Jacobi differential equation

(1 z2)w" + a (a- I b+2)z]u)'+ n(n+ a+ b- f -1)w=0 (7-49)

"The function defined in eq. (6-89) is identical to il(1. n).

225


is a particular Riemann-Papperitz equation with singular points at z=:.+:1, 00.It is readily established that in this case

I 1 1 or

(w= P 0 0 na b n+ a+ b+1

(7-50)

This equation can be put into canonical form by the change of independentvariable t= (1 z)/2. Thus,

0 1 00

0w P( 0

a b n+ a+ b+1Therefore, when a is not a negative integer, a particular solution wl to theJacobi differential equation is

wi= F n+ a+ b+1; a+1; 12z) (7-51)

Equation (7-51) satisfies the differential equation within the circle 11 zi < 2for all values of the parameter n. But if n is a nonnegative integer, the hyper-geometric series terminates after a finite number of terms. For such values ofn, the Jacobi polynomial is defined as

(

n! 2

a+1),, F n+ a+ b+1; a+ 1,12-11 for n=0, 1, 2, . .

(7-52)

A convenient expression for these polynomials is provided by the Rodriguesformula

(-1)n ,pV,b)(z)= [(1z)"l-no+z,b+n, (7 -53)

226


This formula was obtained for the special case of Legendre polynomials (in-troduced below) by Olinde Rodrigues in 1816. In order to derive this result,we use equation (7-34) together with the Jacobi identity (7-47) to show that

t b 6,1147, [ (1 b+nta+n]

t)bdtn

[ta+nF (a+1,-n-ka+1;0]

= (a+1)n(1 t)-4' (a+ l+n,-n b; a+ 1; t) (7-54)

But it follows from equation (7-43) that

(1-t)-T(a-1-1-1-n,-n-b;a+1;t)=F (-n,a+l-Fn+b;a+1;t)

Hence, after substituting this into equation (7-54) and making the change ofvariable t= -z)/2 in the resulting expression, we find, upon comparing theresult with definition (7-52), that the Rodrigues formula (7-53) holds.

A number of important equations which have been extensively studiedare special cases of Jacobi's equation. Thus, when a = b, equation (7-49)

reduces to the ultraspherical differential equation

(1-z2) w"-2 (a+1)zwi-En (n+2a-1-1)w=0 (7-55)

A solution to this equation is given by equation (7-51) with a= b. But,when n is a nonnegative integer and the series terminates,89 it is customary touse a different normalization factor from the one used in equation (7-52) andto express the solution in terms of the Gegenbauer polynomial

C(2oc),,

:(z) =( a ±

Ri[u (1/2)," (I/2)1(z) (7-56)

89 The polynomial obtained directly from eq. (7-52) with a = I) is sometimes called the ultrasphericalpolynomial.

227


where we have put a= a+ (1/2).The Tschebychev equation

(1 z2) w" zw' + n2w = 0 (7-57)

is, in turn, a special case of the ultraspherical equation (7-55) with a=1/2. One solution to this equation is therefore given by equation (7-51)

with a= b= 1/2. However, when n is a nonnegative integer, it is againcustomary to change the normalization and define the Tschebychev polynomial(of the first kind) Tn in terms of the Jacobi polynomial (or ultraspherical poly-nomial) by 9°

!

Tn(z)n p( -112,- 112)(z)1

()ni Another important special case of the ultraspherical equation occurswhen we put a= 0 to obtain Legendre's equation

(1 z2) w"-2zw' + n (n+ 1) w =0 (7-58)

One solution to this equation is given by equation (7-51) with a= b= O.Thus, we define the Legendre function of the first kind Pn by

Pn(z)=F ( n,n+1,1,1?) (7-59)

And since this function is a polynomial when n is a nonnegative integer, it isthen called the Legendre polynomial. Thus, the Legendre polynomial is a Jacobipolynomial with a = b= O.

A second (suitably normalized) linearly independent solution to equation(7-58) about z=1 is called the Legendre function of the second kind and isdenoted by Qn. If n is an integer, Qn will involve logarithmic terms.

Notice that Legendre's differential equation (7-58) (and, in fact, more gen-erally, the ultraspherical differential equation (7-55)) is invariant when z isreplaced by z. This suggests that we transform equation (7-58) by introducing

9° It is not hard to show that 7'. satisfies the relation 7'. (cos 0)= cos n 0, and this relation is often used todefinethis polynomial.

228

R!EMANN- PAPPERITZ AND HYPERGEOMETRIC EQUATIONS

the new independent variable t= z2. The equation then becomes

dw 1201 t) d2 w(1-30 7-ft + -2- n(n+ 1)w -- 0

But this is again a hypergeometric equation whose parameters a, B, and y arenow (1/2 )n, (1/2) (n+ 1) , and 1/2, respectively. And since y is not an inte-ger, we can conclude from the results of the preceding section that this equationhas two linearly independent solutions wi and w2 about t= 0 which are given byequations (7-35) and (7-39), respectively. Thus,

w1=-- F

to2= zF

n n +1 1. \\ 2' 2 2'z)

( n+1 n+2 3 \2 ' 2 ' 2' )

They converge at least within the circle < 1. When n is a nonnegativeinteger, the series for w1 will terminate for even values of n and that for w2will terminate for odd values of n. However, the polynomials obtained in thismanner must be identical, to within a constant factor, with the Legendrepolynomials P n(z).

The associated Legendre equation

(1 z2) w" 2zu" [n (n+ 1)m2

1 z2i w=13 (7-62)

reduces to Legendre's equation when m= 0. It is a Riemann-Papperitz equa-tion for all values of m. Its P-symbol is

w=PC1 1 00

(1/2)m (1/2) m n(1/2) m (1/2) m n + 1

But upon transforming this to normal form, we find that

229


0 0

w= (z2 1)11O 0P mn (1 z)/2

m m m+n+1If we let u be the transformed dependent variable, then

and

w= (z2 1)ml2u

0 00

u=P 0 0 m n (1 z)/2)

rr- m m+n+1

1 00

=P 0 mnm m+n+1

(7-63)

Hence, u satisfies the equation

(1 z2) u " -2 (m+ 1) zu' + (nm) (n+ m+ 1) u =0 (7-64)

But when Legendre's equation (7-58) is differentiated m times, we obtain

dm+2w dm+iw dmw(1 z2)den+2

2 (m+ 1) zdzm+1

+ (nm) (n+ m+ 1) dzm=0(7-65)

Since 13, and Qn are two linearly independent solutions of equation(7-58) about z=1, we see upon comparing equations (7-63) , (7-64), and(7-65) that

Pn (Z) = (Z2 ."/2 d ria 137n L- (z) J

230

and


d"0 (z) = (z2 1)11112 [Qn(Z)]

are two linearly independent solutions to equation (7-62) about z= 1. Theyare called the associated Legendre functions of the first and second kind,respectively.

231 /23,2-

CHAPTER 8

Confluent Hypergeometric Equation and

Confluence of Singularities in

Riemann-Papperitz Equation

Having studied equations whose only singular points are regular singularpoints, it is natural to approach the study of equations with irregular singularpoints by applying a limiting process wherein two or more regular singularpoints of an equation are allowed to approach one another. The process,whereby two or more singular points in any linear differential equation areallowed to come together in such a way that at least one of the correspondingexponents becomes infinite, is called confluence, provided the limiting form ofthe differential equation elxists.

8.1 CONFLUENCE OF SINGULARITIES IN RIEMANN-PAPPERITZ EQUATION

We shall apply this process to the Riemann-Papperitz equation whosesingular points are located at 0, b, and 00 by letting the regular singular pointb approach the regular singular point at 00. Now if we are going to allow theexponents at the points b and 00 to become infinite, we must prescribe themanner in which they approach infinity. This can be done by letting theseexponents be functions of b whose values approach infinity as 00. If wesuppose that the exponents are polynomials in b, it is not hard to show that therequirement that the limiting-form of the differential equation exist impliesthat the exponents be, at most, linear in b. Now equation (7-8) is the geveralRiemann-Papperitz equation with one of its singular points at oo. If we require

233


that the two singular points in the finite part of the plane are at 0 and b and ifwe suppose the exponents at b are 13;+ b[3; and Pi" + b/32" and those at co arey; + by and y;' + by;', this equation becomes

d2w± r 1 a' a" 1 p; 13;' b (g+ f3;' ) 1 dwdz2 L z z b j dz

[

a' a" b (J3'1+ bc(g +' b(y; + bYZ)(Y;'+ b)4' bz)z-0b z

(8-1)

But if the coefficient of w is to remain finite as b co, we must require that

Y;Y'2' = 0

Then upon taking the limit in equation (8-1), we obtain the equation

d2

dz

w2+( 14+ ig;' +1

a' a" \ dwdz

where we have put

(8-2)

[a'a" (a' + a" p;-F ot,z+ f3A w = 0 (8-3)

z2

YiY"-F (1 P;))% (a +Kb%

This equation has a regular singular point at z=--- 0 and an irregular singularpoint at z = co.

234

CONFLUENT HYPERGEOMETRIC EQUATION

8.2 TRANSFORMATION TO KUMMER'S CONFLUENT HYPERGEOMETRIC EQUA-TION

Making the change of dependent variable

w=zae-nzu

transforms equation (8-3) into the equation

dz d2 u + [y 132')z]

ttotitt= 0

dz2 dz

where we have put

(8-4)

y=l+a"a' (8-5)

And, if91 g' # g, the change of independent variable

t= (p;' f3Dz (8-6)

leads 3o Kummer's confluent hypergeometric equation

d2u dut dt--+

(yt) au.= 02 dt

(8-7)

where a= aii(gThus, once the solution to equation (8-7) is known, we can find the solu-

tions to any equation of the form (8-3).

8.3 SOLUTIONS TO KUMMER'S EQUATION: CONFLUENT HYPERGEOMETRICFUNCTIONS

We shall, therefore, seek a solution to Kummer's equation about the regu-lar singular point t= O. Just as in the case of the hypergeometric equation of

in If 02' =0', the change of independent variable z= (r1/16m) and the change of dependent variable u =e--"'"v

lead to the equation r(d2v/dr2) + (2y-1 7) (dvIch) [ (2y 1 )/2]v = 0, which is a special case of eq. (8 -7).

235

488-942 0 - 73 - 16


Gauss, the characteristic exponents at this point are 0 and 1- y. We againsuppose first that y is neither zero nor a negative integer. Then equation (8-7)

00

has a solution of the form u1= ant", which corresponds to the exponent

p= O. And we find, in the usual way, that

Ean(n 1 + y)nta-1 E aft_i(n -1+ ct)t"-1=0n=0 n=1

Therefore, ao is arbitrary and the recurrence relation is

(n-1±a)an= n(n -1+y)

which can be solved to obtain

an-1 forn =l 2. . .

ICY)"n

(('O.n a°

for n=1, . .

Hence, by using the generalized hypergeometric notation, the solution u1can be expressed in the form

where the function

ui = ao (a; y; t) (8-8)

(a; y; t) = ( a). t"nt-O 11!(y)"

(8-9)

is called the Pochhammer-Barnes confluent hypergeometric function. Since theonly singularities of equation (8-7) are at t = 0 and t co. the solution (8-9)must be an entire function.

Other commonly used notations for iFi (a; y; t) are ill(a; y; t) and (1)(a;y; t). A complete discussion of confluent hypergeometric functions is given inreference 28.

236


Notice that by putting t = zip in the hypergeometric equation of Gauss(eq. (7-36)), we obtain the equation

z (1 --z) (pu r(a+P+1)

zi du au=0+ [V 113 dz2 /3 dz

And upon taking the limit /3 00 in this equation, we arrive at the confluenthypergeometric equation (8-7). On the other hand, by putting t = zip in thehypergeometric function of Gauss (defined in eq. (7-33)), we obtain

F (a, p, y, ri(7)(;) )!,):..

And since

(0) nurn z13-, P"

we see that at least formally

z ,lim F (a, 13; 'Y; ita; y; ,Z)0-**,

In a similar way, if y is not a positive integer, consideration of the series corre-sponding to the exponent 1y shows that

u2= 1F1(a-1-1y; 2 y; t) (8-10)

is also a solution to equation (8-12)Thus, if y is not an integer, equations (8-9) and (8-10) provide a funda-

mental set of solutions to equation (8-7). But if y is equal to 1, these two solu-tions are identical. And if y is an integer other than 1, one of the hypergeometricfunctions in equations (8-9) and (8-10) is undefined and the other one providesa solution to the differential equation. However, when y is a positive integer anda is not a positive integer smaller than y, the other linearly independentsolution is given by

237


n!(1 y)n+1u= IF, (a; y; t) in t t(n +1)ft=0 (1 a).+1

(a)n+"S'(y)nn!

[H(a, n) H(y, n) H(1, nfltn?fr.;

where the finite sum is to be omitted when y= 1 and H(a, n) is defined insection 7.6. A corresponding result holds when y is a nonpositive integer, andthe remaining cases should be treated individually by the methods of chapter 6.

8.4 WHITTAKER'S FORM OF HYPERGEOMETRIC EQUATION

There is another standard form into which equation (8-7) (and, therefore,as a consequence, eq. (8-3)) can be transformed. To obtain this equation, wemake the change of variable

Tv= t(1/2)ye-o/2)tu

in equation (8-7) to obtain the equation

&TV [1 /1 1 1

dt2 7 7 a) -71

In order to introduce standard notation, we put

and

1k=-2- ya

1m=.2 (y-1)

to get Whittaker's confluent hypergeometric equation

238

(8-11)

W =0 (8-12)

(8-13)

(8-14)


2W+_4 4/2

m

dt2

)W = 0\ (8-15)

It follows from equations (8-8), (8-10), (8-11), (8-13), and (8-14) that if2m is not an integer

tm+012)e-Wot 1F 1 m k; 2m + 1; t)Mk, m(t) (8-16)

and M k,m(t) are a fundamental set of solutions to equation (8-15) in thedomain 0 < It' < co.

The notation Mk, m is used in reference 25. The Whittaker's functionWk, m is defined by

I'(- 2m) r(2m)Wk, ni(t) Mk, m(t) Mk m(t)

I'(- m k +2

k +2

Since Whittaker's equation (8 -15) is unaltered if we replace k by kand t and t, it follows that

W=M-kon(-t) and W=M-k,-m(-t) (8-17)

are also solutions to equation (8-15) in the region 0 < Id < co. And since thisequation can have, at most, two linearly independent solutions, there must beconstants Cl and C2, such that

Mk,m(t)= C1Mk,m(t) C2Mk,m(t) (8-18)

provided 2m is not an integer. But equations (8-9) and (8-16) show thatMk, m(t) and Mk,m( t) behave like tm-F(1/2), that Mk, m behaves like t-m+(1/2)in the neighborhood of t= 0, and that

239


1 1lirm

teom+0/2/3 ',n(t) = lim tm+(1/2) m(t) = 1

r-*0

Hence, equation (8-18) can hold only if C2= 0 and C1= (-1)""-(1/2). Therefore,

Al_k, m(-1) = ( 1)".4-(1 /2)A1k, m(t)

But substituting equation (8-16) in this equation shows that

1 1et

2+ k + 1+ 2m; = iFie k+ m; 1 +2m; t)

And upon using equations (8-13) and (8-14) to express k and m in terms ofa and y, we obtain Kummer's first formula

IF 1 (a; y; t) = et iF (y a; y; t) (8-19)

This equation holds even when y is a positive integer, for in this case, NQ.!could have replaced Mk. m by the appropriate logarithmic solution andcarried through the proof in the same way.

8.5 LAGUERRE POLYNOMIALS

If the parameter a in equation (8-7) is zero or a negative integer, thesolution 1F1 (a; y; t) (defined in eq. (8-9)) terminates after a finite numberof terms and becomes a polynomial. This polynomial differs from the gen-eralized Laguerre polynomial

(t) = + 1F1( n; 1 +b; t) for n= 0, 1,2, . . . (8-20)n!

b) n

only by a normalizing factor. When b= 0 in equation (8-20), we obtain thesimple Laguerre polynomial L7, defined by

= 1F1( n; 1; t)

Equations (8-9) and (8-20) show that

240


n)i(l+b)dj.fro= j!(1+b);n!

And since (n)iln!= (-1)4(n j)!

and

40(0 = (1+b)(t)j(1+ b)ii4n j)!

1 =0 (302(n-D!

8.6 BESSEL'S EQUATION

An important special case of equation (8-3) occurs when

a" =--- a'=p

The equation

al=2i(a"+12 2

z2w " zw (z2 p2 ) w = 0

(8-21)

(8-22)

(8-23)

(8-24)

obtained in this manner is called Bessel's equation. When p is not a negativeinteger, the methods of chapter 6 can be applied in the usual way to show that(see section 7.5)

1) kz2k + P Z2( 1+ p; )E 22kkul+p)kk=0(8-25)

241


is a solution to equation (8-24) about the regular singular point z= 0. However,it is conventional to multiply this solution by the normalization factor[2PF(1 +p)}---' and to use equation (5-35) to eliminate the generalized factorialfunction in the denominator. When this is done, we obtain the suitable nor-malized solution

where

wi=zip (8-26)

P(Z),;(._-1)k z \2k+P

k2d=ok!r(1 + k+ p) 2 ) (8-27)

is the Bessel's function of the first kind of order p. Notice that this function isdefined even when p is a negative integer. And since z== 0 is the only finitesingular point of the differential equation, the series must converge in theentire plane. Although Bessell functions were used by both Leonard Euler andDaniel Bernoulli before Besse' was born, the German astronomer FredrickWilhelm Bessel was the first to make a detailed study of them.

Since the function in equation (8-25) differs from Jp only by the normaliza-tion factor [2PF(1+ p)]-1 when p is not a negative integer, it follows that

z2)J p(z)

(i)PF(l+p) oF ; 1+ p: - -4 for p -1, -2, . . . (8-28)

On the other hand, we know that equation (8-24) can be transformed intoKummer's confluent hypergeometric equation (8-7) by using the change orvariables given by equations (8-4) to (8-6).

The solution Jj, can, therefore, be expressed in terms of a linear combina-tion of solutions of equation (8-7). Thus, if p is not an integer, it follows fromequations (8-4) to (8-6), (8-8), (8-10), and (8-23) that

Jp(z) AzPe-iz 1F1 ,1+ 2p; 2iz)+ Bz-Pe-iz 1F1 p; 1-2p; 2iz)

242


But upon comparing the behavior of these functions at z=0, we find, whenp is not an integer, that this equation can only hold if B = 0 and thatA= [2Pr(l +p)]-'. Hence,

zPe-izjp(z)-2Pr(l+p) 1F1 (p+-2 ;

1 +2p; 2iz

Comparing this with equation (8-28) shows that

2

OF1 (; 1+ P;1 ; 1+2p; 2iz)

Then by putting = iz and a= p+ (112) , we obtain Kummer's second formula

1

'10F1 ( ; ot+-

2-- 442) = e-4 1F1 (a; 2a; 24)

When p is a negative integer, there is a close relationship between Jpand J-p. Thus, suppose p=n for n=1. 2, . . . Then since the T.-functionhas poles at each of the nonpositive integers, all the terms in the sum (8-27)will vanish for k < n; and we obtain

)kn(Z) k!F (1

(1-

k n) (2

Nut this becomes, upon shift of index from k to k n,

1)k+ n 2k+nJn(Z) (k+ n) !f(1 +k) 2)


J_,(z)-= (-1) n. ) for n=1, 2, . . (8-29)

243


Since iv' =,/p is a solution to equation (8-29) for all values of p and since theparameter p enters equation (8-24) only as p2, the function

w2=J_p(z) (8-30)

must also be a solution to this equation for all values of p. When pis an integer,equation (8-29) shows that the solutions wi and w2 are not linearly independent.However, for any nonintegral value of p, consideration of the behavior of thefunctions Jp and J_p in the neighborhood of z= 0 shows that wi and w2 consti-tute a fundamental set of solutions. In order to obtain a second linearly independ-ent solution to equation (8-24) when p is an integer, notice that for any valueof p

Z2,1p " Z + (Z2 p2)Jp---= 0(8-31)

z2f Lip+ zJi_p+

Since the series (8-27) is uniformly convergent, it can be shown that Jp, J_p,and their derivatives with respect to z are all differentiable with respect to theparameter p. Hence, upon differentiating equations (8-31) with respect to pand subtracting the result, we obtain

z2,29"; +z /p+ (z2 p2),,fp = 2p [.1P- (- 1 )7g _p] (8-32)

where we have put

aJp _pp (-1)71

a p p

By taking the limit as p--> n in equation (8-32) and using equation (8-29),we find that the Bessel function, of the second kind

I n(Z) 1 11111 raj )(Z) ( 1)n al-1(Z)]Tr L ap J

(8-33)

is a solution to equation (8-24) with p= n. But upon substituting equation (8-27)into this expression and noting that

244


lim1 dr (-1)k-1-1k! for k= 0, 1, 2, . .

l2 (z) dz

it follows after some algebraic manipulation, which we shall omit here, that

(h- k-1)! (1 2k-nYn(z) = (y+ In 0J,,(z)- n2-1

7r k! 2

1 (-1)kTr z\12k+n

it 14,0 k!(n+ k)! (Hk±nn+k)(8-34)

where the finite sum is to be omitted if n= 0, Hk is defined by equation (6-89),and y is Euler's constant defined by

y= lim (Hk In k) = 0.5772156649 . . (8-35)

It is easy to see from equations (8-27) and (8-34) that Y. and h are a funda-mental set of solutions.

It is convenient to extend the definition (8-33) of Yn to nonintegral values ofp. This is now done by putting

Yp(2)jp(z) cos p7r -AZ)

sin pir (8-36)

when p is not an integer. Then Vp and j p are a fundamental set of solutions.And both the numerator and denominator of this expression vanish whenp is an integer. But by using L'Hospital's rule, we find that for n= 0, 1, 2, . . .

aip fa-pcos Tip- IT J p sin pIrpj cos per -J-P_ . ap aplim Yp (z) = lim hrnp-n p-n sin pir pn IT cos prr

1 . 1 [af= kJ_= limir fr-,n a p cos pir ap IT p--, cap ap

And comparing this with equation (8-33) shows that definition (8-36) is indeedan extension of definition (8-33).

245


For large values of z with arg zi < 7r, the Bessel functions jp and nbehave like

Jp(z) (1)1" cos (z PIT-4-

7r7rz 2

7r 7rYp(z) i+z ) "2 sin ( z

2 -4-

(8-37)

However, it is sometimes convenient to have solutions to Bessel's equationwhich tend to zero exponentially as Izi > oo in the entire half planeLnz > 0or in the half plane Sw z < 0. But since equations (8-37) show that for large z

Jp(z) ± iYp(z) --- (-2 ) 112e-±i[z-Onr/2)-(1114)]

Irz

we see this property is possessed by functions Jn±: iYp. For this reason, theHankel functions of the first and second kind Hu) and FP), respectively, aredefined by

11(71,)(z)=-- p(Z) iYp(z)

1-1(2) (z) = p(z) iY p(z)

It is not hard to show that these solutions are a fundamental set.In a number of applications, we encounter the equation

z2wH + zw' (z2+ p2)w = 0 (8-38)

which is obtained from Bessel's equation by replacing z by iz. Although Jp(iz)and Y p(iz) are a fundamental set of solutions of equation (8-38), it is convenientto introduce new functions which are real for real values of z (at least for realvalues of p). To this end the modified Bessel functions of the first and thirdkind 1p and Kp, respectively, are defined by

Ip(z) -7 e-i(P42)jp(iz)

246


K p(z) =74 ei(Pn12) (iz) = 2L ei(P742)Up(iz) + iY p(iz)]

There are a number of useful relations between the Bessel functions andtheir derivatives. Thus, it follows from equation (8-27) and equation (5-31)that

1) kz2k+2pZPJ p(Z) =

k22k+ Pk! (k p)r (k p)=0

But upon differentiating both sides with respect to z, we find that

w I_ 1)kz2k+2p-1Tiz (ZPJ p(Z)) =

22k+P-4-1k! r(k + p)= p -1 (Z)k=0

In a similar way it can be shown that

dz z-PJ p(z) p+1(z)

(8-39)

(8-40)

After carrying out the differentiations in equations (8-39) and (8-40), we obtain

yZ

p Z./ p- Pip (8-41)dz

zdz

p

And by adding and subtracting these equations, we find

and

d2

dz-

p -Jp+1

(8-42)

(8-43)

2pfp= zip-1 + (8-44)

247


The formulas (8-39) to (8-44) are very useful when dealing with Bessel func-tions. They also apply to the Bessel functions Yp, 141), and I/V). However,they must be modified slightly for the functions /p and Kp (due to the presenceof iz).

A much more complete treatment of Bessel functions can be found inreference 29.

8.7 WEBER'S EQUATION

Another equation, which has been extensively studied, can be obtainedfrom equation (8-3) by putting

to get

a"=0 1

14 = 13;=4,

zw"+2 1

12 +n -4z)

and then making the change of independent variable

to obtain Weber's equation

=

(+ od2 k," 2 4

=-4, (8-45)

(8-4.6)

(8-47)

(8-48)

This equation was first studied by Hermite (1864) and then by Weber. Itssolutions are called parabolic cylinder functions or Weber functions.

Since equation (8-46) can be transformed into the confluent hypergeo-metric equation (8-7) by the change of variables (8-4) and (8-6), equation(8-48) can be transformed into equation (8-7) by the change of variables (8-4)to (8-6) and (8-47). Hence, the solutions to equation (8-48) can be expressed interms of the solutions to equation (8-7). Therefore, use of equations (8-4) to(8-6), (8-8), (8-10), (8-45), and (8-47) shows that a fundamental set of solu-tions to Weber's equation (wl, w2) is given by

248


n'

1

21wi.e--014)42

1F

1 2 - - 42)' 2

n-1 3 1 r2\e-0/4X2421Fi.2 ' 2' -2 /

(8-49)

(8-50)

Notice that if n is a positive even integer, the hypergeometric function inequation (8-49) terminates and becomes a generalized Laguerre polynomial;while if n is a positive odd integer, the hypergeometric function in equation(8-50) becomes a generalized Laguerre polynomial. Thus, it follows from equa-tions (8-20), (8-49), and (8-50) that, if n is a positive integer, equation (8-48)has one solution w3 of the form

W3= e"14)C2!In() for n=1,2, . (8-51)

where this the nth Hermite polynomial, which is defined in terms of the gen-eralized Laguerre polynomials by

112k = (-1)1c22k kl(k-1 /2) (42)

k=0,1,2,. . . (8-52)ii2k+1 (1) = (- 1 ),V22k+Ik 1.44(1/2)(47)

8.8 OTHER EQUATIONS STUDIED

Although we have been able to give a complete presentation of the gen-eral theory of the solutions to second-order linear equations about regular sin-gular points, apart from relatively trivial cases, only the solutions to thoseequations which can be transformed into the hypergeometric equations ofGauss and of Kummer have been exhaustively studied. The equations ofLarni and Mathieu have also been studied, but only a small amount of knowl-edge of the functions defined by these equations has been obtained. TheMathieu equation is

du;+ (a+ k2 cos2 z)w= 0

where a and k are parameters, and the Lame equation is

249


d2wdz2

+ [77+ n(n+ 1)msn3(zint)]w=0

where sn(zlm) is the sine amplitude elliptic function of modulus m, and v, n,and m are parameters. When n=2, its solutions can be expressed in terms ofelliptic functions. For more information about these equations, the reader isreferred to reference 3. An extremely complete discussion of Mathieu's equa-tion with many applications is given in reference 30.

All the equations which have been discussed in the last two chapters canbe derived from a single equation with six distinct regular singular points (withthe characteristic exponents at each singular point differing by 1/2) by allowingthese singular points to coalesce. The coalescence of any two such points is aregular singular point whose exponents differ by an arbitrary amount. Thecoalescence of three or more such points will result in an irregular singularpoint. The reader is referred to reference 3, chapter 20, for a more detailedstudy of these matters.

250

CHAPTER 9

Solutions Near irregular Singular Points:

Asymptotic Expansions

9.1 GENERAL CHARACTER OF SOLUTIONS

Let zo be an isolated singular point of the differential equation

(121 dwdz2 +p(z) dz

q(z)w=0 (9-1)

We have shown in chapter 6 that it is always possible to construct a funda-mental set of solutions to this equation on a punctured circular region aboutzo whenever zo is a regular singular point. And these solutions can be expressedin terms of certain convergent power series whose coefficients can be calculatedsuccessively. This occurs because the functions f, (z) and f2(z) in the canonicalbasis (6-20) can have, at most, poles at zo. However, this will not be the casewhen the point zo is an irregular singular point of equation (9-1) since at leastone of the two functions ft (z) and f2(z) will then have an essential singularityat zo. This function will therefore have a series expansion about zo of the form(6-21) in which infinitely many negative powers of zzo are present. And thecoefficients of this series are determined by an infinite set of equations whichcannot be solved recursively. For this reason we do not have a convenientmethod of finding the corresponding solution. In addition, the characteristicexponents are determined by a transcendental equation (which usually cannotbe solved explicitly) instead of by an algebraic equation. Finally, the seriesonce found are usually slowly convergent.

251

488-942 0 - 73 - 17


9.2 FORMAL POWER SERIES

Even if zo is an irregular singular point, it may still happen that the functionft (z) in the canonical basis (6-20) has, at most, a pole at zo, in which case itwould still be possible to recursively construct 92 one solution about this point.And this solution would be ,t-,f the form

tv= (z-zo)Pf(z) (9-2)

where f(z) is analytic in a neighborhood of zo,, and therefore has a convergentpower series expansion

00

f(z) = an(z-zo)n (9-3)n=o

about zo. We assume, without loss of generality, that the exponent p is adjustedto make ao 0 0 and therefore to make

f(zo) 0 0 (9-4)

A second solution could then be found, at least in principle, from equation(6-63).

When p(z) has an essential singularity at zo, equation (9-1) can possessa solution of the form (9-2) only when q(z) is equal to zero or nas an essentialsingularity at zo. But suppose that p(z) has a pole at zo, say of order m. Thus,

p (z) = (z- zo) inP (z) (9-5)

where P(z) is analytic at z= zo. Now if equation (9-1) possesses a regular solu-tion of the form (9-2), this solution together with (9-5) can be substituted intoequation (9-1) to obtain

2 If it were the function fo(z) which had the pole at zo and the function jr(z) which had the essential singularity,then either we could (when a =0) change the notation so that the function fo(z) had the pole at zo or (when a # 0) thesolution wo would depend upon fo (z) (through to (z)) and could therefore not be constructed recursively.

93 Recall (ch. 6) that a solution of this type is called a regular solution.

252

ASYMPTOTIC EXPANSIONS

q(z) 2p_.,f'(z) p(p-l)f (z) z f (z) :0)2

[I ft (Z)(z zo) f (z) (z zo)m+] P(z) (9-6)

And since f(zo) 0 0, it follows that the functions fulf, f'lf, and P are analyticat s = so.

First, suppose that m is either zero or 1. Then equation (9-6) shows thatq(z) has, at most, a pole of order two. We therefore conclude that when p(z)has, at most, a simple pole at zo, equation (9-1) will possess a regular solutionof the form (9-2) if, and only if, zo is a regular singular point.

Now, suppose that m > 1. Then equation (9-6) shows that q(z) has, at most,a pole of order m + 1. Hence, we conclude that when p(z) has a pole of orderm > 1 at a point zo, equation (9-1) can possess a regular solution about thispoint only if q(z) has, at most, a pole of order m+ 1 at so.

The corresponding result for the case where the point zo is at 00 can beobtained in the usual way by applying these results at the origin of the 4-planeto the coefficients of the transformed equation given by equations (6-28) and(6-29). This leads to the conclusion that if p(z) has a pole of order m

about s=00, equation (9-1) possesses a solution about this point of the form

ZP E"..n=0 z

only if the highest power of z which occurs in the Laurent series expansion ofq(s) about z = is m 1.

Now suppose that so is a finite point, p(z) has a pole of order m at zo, andequation (9-1) possesses a solution of the form (9-2) about so. Then, q(z) mustbe of the form

q(z) (z zo)-711-'Q(z) (9-7)

'm Recall that we say that a function has a pole of order zero at a point if it is analytic at that point.

253


where Q(z) is analytic at z= zo. This function Q(z) and the function P(z)which appears in the representation (9-5) of p(z) can be expanded about zoin the convergent power series

P(z) = Po(z zo)nn =0

Q(z) qn(ZZO)nn=0

(9-8)

where po 0 0.We can now proceed much as we did in chapter 6 for the case of a regular

singular point by substituting equations (9-2), (9-3), (9-5), (9-7), and (9-8)with m > 1 into equation (9-1) and then collecting the coefficients of likepowers of z zo. When this procedure is carried out, we find first that theindicial equation is not quadruic (as in the case of a regular singular point)but that it is an equation of the first degree which is

PP0 + qo = 0 (9-9)

This is consistent with the fact that, at an irregular singular point, at mostone solution of the form (9-2) can occur.

Next, the coefficients of the series (9-3) can still be calculated recursively,but the resulting series will now either terminate after a finite number of termsor else it will diverge for all zol > 0 (ref. 3 p. 421). Thus, a regular solutionwill only exist in the exceptional case when the series (9-3) contains a finitenumber of terms.

For example, the equation

+ 2 (zz

a)w,

3-. z3 (a +

4z) w =0

has an irregular singular point at z = 0. And it has a series solution about thispoint given by

254


which terminates after two terms.Even though it is possible to calculate as many terms as desired when the

series does not terminate, the solution obtained in this manner will not haveany meaning in the usual sense since the series does not converge. This shouldnot be taken to mean, however, that this "formal" solution is not useful.

9.3 ASYMPTOTIC CHARACTER OF SOLUTIONS

In order to introduce the concepts which allow the formal solutionsobtained in the preceding section to be utilized, it is convenient to consideran example given by Euler in 1754. Thus, the differential equation

tel 3z + 1 1w = 0w' + z2

z2(9-10)

has an irregular singular point at the origin and a regular singular point atinfinity. Since the order of the pole of the coefficient of w at z = 0 does notexceed the order of the pole of the coefficient of w' at z= 0 by more than 1, weknow that equation (9-10) possesses one (and only one) formal solution aboutz= 0 of the form

CC

w = zp E anznn=0

(9-11)

We can therefore proceed just as in the case of a regular singular point andsubstitute this expansion into equation (9-10), shift the indices to the lowestone present, and collect terms to obtain

E(n oan_izn+p-i E (n + p)anzn+P-i = 0n=1 n=0

Then by equating the coefficients of the various powers of z to zero, we findthat since ao 0 0

p = 0 (9-12)

255


and

(n + Oan-i = (n + p)an for n = 1,2,3, . . . (9-13)

The indicial equation (9-12) is consistent with the general indicial equation(9-9) since, in this case, qo = 0. When equation (9-12) is substituted into therecurrence relation (9-13), we obtain

an= nan-i for n = 1,2,3,. . (9-14)

And by proceeding in the usual way, we conclude from this, that

( 1) nn!ao

Finally, substituting this, together with equation (9-12), into equation (9-11)shows that equation (9-10) possesses the formal solution

w= E (- 1) nnlen=0

(9-15)

A simple application of .the ratio test, however, shows that this seriesdiverges for all z 0 0. Hence, as we have already indicated in the general case,equation (9-15) is not a regular solution. However, we might anticipate that,since equation (9-15) was determined in a formal manner by the differentialequation, it might still, in some sense, represent a solution to that differentialequation.

In order to see what this is, note that equations (5-29) and (5-36) show that

n! =10 e-etndt (9-16)

256


However, when this is substituted into equation (9-15), we find that

w f et(-1)nzntndtn=0

And again, proceeding formally we interchange the order of summation andintegration to obtain

w =-- f e-t ( zt)ndt0 n=0

Finally, summing the geometric series shows that

e-tw = dt

o + zt (9-17)

Now this integral converges uniformly95 for all z in any region R boundedaway from the negative real axis. In fact, differentiating with respect to z andinterchanging the order of integration and differentiation shows that

dw I todz Jo (1 + zt)2

dt (9-18)

"5 This means that for any positive number e, no matter how small, there exists a positive number T(e) which

depends on e but not on z such that I I [e-e/(1 + zt)] dtl< e for all t T(e) and all z in R.

257


e-tte-rFIGURE 9-1.Region of convergence of 10

(1-1-zt)2d,and

Jdt.Joy

(1 zt)2

But since this integral converges uniformly for all values of z in any regionbounded away from the negative real axis (fig. 9-1), we can conclude that thederivative exists and that interchanging the order of integration and differentiation was in fact justified, provided z is not on the negative real axis (ref. 31).

f0Hence, [e-t/(1 +zt)]dt is an analytic function of z everywhere in the z-

plane cut along the negative real axis and is therefore infinitely differentiablein this region.

Upon multiplying equation (9-17) by z and equation (9-18) by z2 andthen adding the result, we get

L 2 + zw= z2 10(1+ zt) 2

dt+z1 e+ zt

d

= e_tz(1+zt) e_ezz2t dt= dt(1+ zt)2 o (1+ zt)2

And after integration by parts, this becomes

258


--tz2 + zw= e

1 +-t ozt o

"1+ zt

dt = 1 w

which shows that w satisfies the equation

z2 w' + (z + 1) w = I

provided z is not on the negative real axis. Since w is infinitely differentiable,we can certainly differentiate this equation with respect to z. But when this isdone we obtain equation (9-10). Thus, the function w given by equation(9-17) provides a perfectly valid solution to equation (9-10). This solutionwas obtained by formally summing the divergent series (9-15), which was, inturn, obtained by formally solving equation (9-10). It is, therefore, natural toask in which sense the divergent series (9-15) represents the integral (9-17)and, therefore, as a consequence, in what sense it represents a solution toequation (9-10). To answer this question, let

sm(z)= E (-1)n,dz. (9-19)n=0

be the mth partial sum of the series (9-15). Now from elementary algebra, weknow that the sum of the finite geometric series is

1 1rwhich becomes, upon setting r=zt,

1 ( zt)m+11 + Zt 1 + Zt 1-4=0

But, upon multiplying both sides by e-t and integrating between 0 and o, wefind that

e-tJo 1+ zt

dt= J Em (-1)n(zt)ne-tdt+Rm(z)

259


where we have put

tm+le-tdtRm(z) = ( z) "1+1 io

1+ zt (9-20)

And since it is always legitimate to interchange the order of integration andsummation for a finite sum, this becomes

r e-tJo 1+ zt.t (-1)nzni tne-tdt= Rm(z)

n=0

Hence, substituting in equations (9-16) and (9-19) shows that

Now since

fo 1 +

e-tzt dt Sm(z) = Rm(z)i

11 + ztl = 11 + xt + iyt1 = V (1 + xt)2 + y2t2

it follows that

1 1 + zt1 = V(1 + xt)2 = 1 1 +xt1 1 fot z= x > 0

And for g17, z= x 0 we find that

(9-21)

11 + zt1 V(xt)2+ (0)2= { (1 + xt)2(xt)2 + y2t2{(1 + xt)2 + (x02] + (y0411/2

1Vy2t2[(1 + xt)2 + (xt)2J

Hence

1 101 1 12Y±c2- 'sin 01V2 V(xt)2+ (0)2 v2

260

for A,z =x<0


where 4) is the argument of z. We have therefore shown that

and

1

ztifor A. z 0

1

+ zt)2 (cosec 01 for A 0; 101 < 7T

It now follows from equations (9-20) and (9-22) that when ,A z 0

x --t

IRm (z) I = tz)"'+' 1 en+ e dt0 1+ zt

t"1-"e-t1+ zt

And equations (9-16) and (9-21) therefore show that

where

dt Izin1+1

iwi(z) -Sm(z) I kr" (m+ 1) ! for 0

r e -1w1(z) 1+ zedto

(9-22)

(9-23)

(9-24)

(9-25)

is the solution to equation (9-10). In a similar way, it follows from equation(9-23) that

Iwi (z) -Sm(z) I Ls. .\/) cosec [di"' (m+ 1) ! for z< 0;10 <

(9-26)

261


Thus, in view of equations (9-15) and (9-19), the inequalitth4 (9-24)and (9-26) show that, provided (A, the argument of z, is smaller in absolutevalue than IT, the error incurred by computing the solution wi(z) by means ofthe divergent series (9-15) goes to zero rapidly as z-) 0. Thus, for any fixed m,any desired accuracy can be obtained by taking Izi sufficiently near zero. Inparticular, the inequality (9-24) shows that, provided 4. z> 0, the error isnever greater in magnitude than the first term omitted in the series.

Now recall that a series a. is said to converge to a sum A if its partialdeo

sums t.de an satisfy the relation

3

14.11 A tmi= °

The series (9-15) certainly does not converge to the solution w1(z) since

lirniwi(z)-Sm(z)1 00 (9-27)

But this series is said to be an asymptotic expansion (which will be definedsubsequently) of wi(ip since, as can be seen from equations (9-24) and(9-26), it has the property that

liW 1 (Z) S (Z)rn

zm-0 for 14)1,-5 7r+e

for any e > 0.

9.4 ASYMPTOTIC EXPANSIONS

(9-28)

Roughly speaking, equation (9-28) shows that the difference betweenthe solution wi(z) and the mth partial sum Sm(z) of the series (9-15) approacheszero more rapidly than zm as z--> 0. Comparing the definition of convergencewith the corresponding condition (9-28) for a series to be asymptotic, showsthat the essential difference between these two concepts can be stated in thefollowing way: If the series is convergent to w1(z) , then z is held fixed and thedifference wi(z) Sm(z) approaches zero as in approaches infinity. If theseries is asymptotic to wl(z), then the number in of terms is held fixed and the

262


difference wi (z) approaches zero (at a specified rate) as z approachessome fixed value, say zo.

There is no reason why a Oven series cannot be both asymptotic and con-CO

vergent at the same time. For example, the geometric series I zn is convergentn=0

to 1/ (1 z) for izI < 1 and it is also an asymptotic expansion of this functionas z -0 0.

When a series is convergent as well as asymptotic, the approximationcan usually be improved at fixed z by taking more and more terms in the par-tial sum. However, it happens more frequently that an asymptotic series isdivergent.96 When this is the case, there is some limiting value to the accuracyof the approximation which can be obtained at each fixed value of z. This limit-ing value is attained when the optimal number of terms is retained in the expan-sion. Thus, l'or the expansion (9-15) with z > 0, the right side of the inequality(9-24) is a, minimum when in 1z1-1. For this value of rn, Sm(z) will providethe best approximation to w, (z).

The theory of asymptotic expansion was initiated by Stieltjes and by Poin-care at the end of the nineteenth century. Before giving the general definitionof an asymptotic expansion, it is convenient to introduce the concept of order.Thus, if 4) and Ii are complex functions of z, we write

(t)= O(P) as z zo

if there exists a finite constant A such that

(z)lim < A

tf1(z)

And we write

cp.= o(tp) as z --o zo

if

lim -4211= 0

"Usually in this carne the terms of the series first decrease rapidly (the more rapidly the closer the independentvariable is to its limiting value) but eventually start to increase again.

263


The symbols 0 and o are called order symbols, and in both cases we say that4 is of order Ir. Thus,

but

Also

but

and, for Pe z > 0,

z4 = 0(z5) as z-)

z4 =o(z5) as z-* 00

z4= 0(4z4+3z3) as z--4 00

z4 o(4z4+3z3) as z--0 00

z5=o(z4) asz-00

,z4 o(z5) asz-+0

e-z=o(za) as -0 00

for any constant a.If f (z) is an analytic function which has a pole of order m at zo, then

f(z)=0((z-zo)-m) and f(z) o((z- zo)-m) as z-÷zo

It is common practice to denote any sequence, say al, a2, a3, . . by{ai}. A sequence of functions {4j(z)} is said to be an asymptotic sequence as

zo if for each integer j, y51+1=0(0.0 as z-* zo. Thus, {(z zo)i} is an asymp-mnic sequence as z-> zo; {z--1} is as asymptotic sequence as z -> 00; and {e-Jz} isan asymptotic sequence as z--* co, provided A z> 0.

264


Note that equation (9-28) can be written in terms of the order symbol o as

wlsm=o(e) as z -+ 0

We are, therefore, led to the following definition: Let {4:1);(z)} be an asymptoticsequence as z--0 zo. Then the formal series

ajcf)j(z) (9-29)

with partial s; ms Sm(z) = ajOi(z) is called an asymptotic series. TheJ.0

asymptotic series (9-29) is said to be an asymptotic expansion to m termsof a function f(z) if

and we write

f(z)Sm(z)=o(4),,i) as z--0 zo (9-30)

fiz) aid)j(z)

If equation (9-30) holds for every positive integer m, we simply say thatequation (9-29) is an asymptotic expansion off (z) and we write

fiz) aicAj(z)J =0

Thus, in the preceding example, we have shown that

fox -----i-e+-1(zktE (_ 1)n /11Znn=0

This series is typical in that most asymptotic series encc intered in practiceinvolve powers or inverse powers of z. Such series are called asymptotic powerseries.

Once a particular asymptotic sequence has been specified, any givenfunction will have only one asymptotic expansion in terms of this sequence.

255


In this sense the asymptotic expansion is unique. However, two differentfunctions may have the same asymptotic expansion. Thus, if

f(z) E anZ-nn=0,

it is not hard to verify that

as z 00

f(z)+e -z E anz-n as z 00

n=0

for A 3 > 0.It can be shown that if f(z) E anchn(z) and g(z) E bnOn(z), then

af(z)+ 13g(z) E (actn+ (31),)41,1(z)

for any complex constants a and 13. It is generally possible to integrate asymp-totic expansions term by term, but differentiation is not always permissible(ref. 32) .

Asymptotic power series can be manipulated in much the same way asconvergent power series. Let the function f(z) be a single-valued analyticfunction at every finite point z with lz I > R. Iff(z) has the asymptotic expansion

1(z) E a nz-" as z 00n=0

and this expansion is valid for all values of the argument of z, then f(z) mustbe bounded as z > co. Hence, it cannot possess a pole or an essential singu-larity at z=00. Therefore, an analytic function cannot possess a single asymptoticexpansion in inverse powers of z as z--÷ co which is valid for all values of theargument of z unless it is analytic at infinity. Thus, asymptotic power seriesare frequently valid only over a given sector of the complex plane, and adifferent asymptotic representation must be used outside this sector. Thisis closely related to the so-called Stokes phenomenon which we shall encountersubsequently.

If a function f(z) is analytic at a point zo and, therefore, possesses a con-vergent power series about zo, it is not hard to see that the series must alsobe an asymptotic expansion off (z) as z zo (ref. 32, p. 22).

266


9.5 NORMAL SOLUTIONS

Let zo he an irregular singular point of the equation

w"- p(z)w' q(z)w=0 (9-31)

and suppose that, if zo is a finite point,

p= 0( (zzo)-m) and q= 0 ((z zo)--(k+o) as z zo

and that, if zo=

p=0(z'n) and o(ik-o) as 7.-+ co

where m and k are integers.We have seen that, when k m (and only in this case), it is always possible

to construct a formal series solution wi (z) of the form

wi(z)= (z z,))P an(zzo)n=0

or of the form

11 if zo is finite (9-32a)

w, (z) = zpE anz-" if zo.= co (9-32b)n=0

which satisfies the differential equation term by term. The coefficients of theseexpansions can be calculated recursively from the differential equation inmuch the same manner as those in the series solutions at regular singularpoints. These expansions will, in general, be divergent. However, as we haveseen by example they actually turn out to be asymptotic expansions of a truesolution to equation (9-31) at the point z---=--zo. They will, therefore, be veryuseful for numerical computation 97 of this solution for z near zo.

The series in equations (9-32a) and (9-32b) will not diverge only if theyterminate after a finite number of terms, in which case these equations willrepresent regular solutions to the differential equation (9-31). A second solu-

97 In fact. asymptotic expansions are quite frequently more suitable for numerical computation than convergentpower series since fewer terms are required to obtain a Oven accuracy.

267

488-942 0 - 73 - 18


tion at the point zo can then always be found from equation (6-63). Thus, exceptin the relatively rare case where k m and the formal power series solutionterminates, it is still necessary to find one or two solutions to equation (9-31)or at least the asymptotic expansions of these solutions about the irregularsingular point z= zo. To this end, we introduce into equation (9-31) the changeof dependent variable

where

SI=

1E COn(Z ZO) 71 if zo is finiten=1

8

w-- eau

n=1tOnZn if zo =

(9-33)

(9-34)

and the coefficients oh, for n= 1, 2, . . s are complex constants. Thenequation (9-31) is transformed into the equation

where

and

u"+ p+(z)u' q+ (z)u= 0

p+ (z) = p (z) + 2S/'

q+ (z)= q(z) +p(z)fl' +Si "+ Cr 2

(9-35)

(9-36)

(9-37)

Now if, when zo is finite, the terms in (z) can be chosen to cancel outenough of the negative powers of z zo which arise from p(z) and q(z) so thatp+ (z) has, at most, a simple pole and q+ (z) has, at most, a pole of order two atzo, then zo will be a regular singular point of equation (9-35). Similar remarks,of course, apply when zo= 00. In either case, then, when this can be done, a

268


fundamental set of solutions can always be found by the methods of chapter 6;and equation (9-33) will then provide a fundamental set of solutions of equa-tion (9-31). But this usually cannot be done. More frequently, it is only pos-sible to adjust the coefficients in SZ so that the resulting equation,(9-35) willpossess a formal solution either of the form (9-32a) (when zo is finite) or of theform (9-32b) (when zo is infinite). Then equation (9-31) will have a formal solu-tion either of the form

W (Z) - efl( (z-zo)P ao(z-ZO)nn=0

or of the form

if zo is finite (9-38a)

w(z)= eO(z)zP E anz-n if zo = (9-38b)n=0

These formal solutions, called normal solutions, either will terminate or theywill diverge in which case they will be asymptotic expansions.

Before discussing this in more detail, it is convenient to introduce thefollowing definition: If the coefficients p(z) and q(z) of the differential equation

w"+ p(z)w' + q(z)w=-- 0 (9-39)

have, at most, poles at zfi, the equation is said to be of finite rank at zo. If zo isa finite point and equation (9-39) is of finite rank at zo, the smallest number rsuch that both

p=0((z-Z0)-(r+1)) and q=0((z- zo)-2(r+ 1)) as z zo

is called the rank of equation (9-39) at zo. The rank of equation (9-39) atz= c is defined to be the smallest number r such that

p= 0 (zr- I) and q 0 (z2(r )) as z--) co

provided the equation is of finite rank at 00. If the equation has a regularsingular point at zo, it is said to he of rank zero.

269


For example, since the coefficients of the equation

satisfy the conditions

p=0(z-')

p 0 o(z-9

ZW"-F w= 6 (9-40)

q= 0(z-91

q o(z-1)as z->

we see that this equation is of rank 1/2 at z = co.Now suppose that equation (9-31) is of rank r > 0 at the point zo. In order

that equation (9-35) possess a formal solution of the form (9-32a) when zo is afinite point, it is necessary that the highest negative power of z- zo in q+ exceedthe highest negative power in p+ by no more than 1. And if equation (9-35) isto possess a formal solution of the form (9-32b) when zo= 00, it is necessarythat the highest power of z in q+ be less than the highest power of z in p+.Suppose that z0=--- oo. An examination of the expression obtained by substitutingequation (9-34) into equations (9-36) and (9-37) shows that q+ will alwayscontain higher powers of z than p+ unless the numbers a and s in equation(9-34) can be chosen to cancel out the coefficients of those powers of z in q+which are larger than, or equal to, those in p+. This can be done only if the rankr of equation (9-31) is an integer. And the maximum value of s necessary toaccomplish this will then always be less than, or equal to, r. The same con-clusion applies when zo is finite. It is usually possible to choose the numberscan and s so that the cancellation is accomplished in more than one way. Inthis case, we obtain two solutions to equation (9-31).

Thus, equation (9-31) will possess formal solutions either of the form(9-38a) or the form (9-386) at the point zo only if the rank r of this equation atzo is an integer. The function 1-1,(z) is then given by equation (9-34) with s r.

Since equation (9-40) is of rank 1/2 at z =00, it does .not possess a normalsolution at this point.

As an example of an equation which does possess such a solution at z =00,consider the equation

w" q(z)w=0 (9-41)

270

with

and

q(z) = E thiz-"n=0

qo 0


(9-42)

(9-43)

We see in view of condition (9-43) that equation (9-41) is of rank 1 at z= co.Hence, we make the change of dependent variable

w en u

where, since s r= 1,

(9-44)

(9-45)

And upon substituting equations (9-44) and (9-45) into equation (9-41) wefind that

u" p+ (z)ui F q+ (z)u= (9-46)

where p+ (z) = 2w and q+ (z) --= go+ qnz-n. The highest power of Zn=

which occurs in both p+ and q+ is zero. Hence, the highest power of z whichoccurs in q+ can be made less than the highest power which occurs in p+ bychoosing col to be a root of the equation

co+q0=0

But the two roots of this equation are 0)1,1= iq1012 and co 1,2.= iqW2. And each ofthese roots leads to a different form of equation (9-46). Thus, we obtain thetwo equations

271


2iqW2u'i+ 4(z) = 0 (9-46a)

2iq,1312uf,± 4(z) u2= 0 (9-46b)

where -4(z)-- qnzn. But we know that these equations possess formalsolutions of the form

and

u,=zpi E anz-"n=0

u2 =z1.2 E bnz-nn=0

(9-47a)

(9-47b)

where pi and p2 are each solutions of linear indicial equations and the coeffi-cients an and bn can be calculated successively from a recurrence relation ofthe usual type.

Thus, for example, upon substituting equation (9-47a) into equation(9-46a), shifting the indices, and collecting coefficients of like powers ofz in the usual way, we obtain

E(Pi -n+2) (Pi-n+1)an-2z-nn=2

+ E z-n [24'12 (p n + 1)(2,4-1 + E qkan_kl = on =1 k=1

or,

And after equating to zero the coefficients of like powers of z, we find that forn=1 (since ao 0 0)

2iq4/2pl+qi=0 (9-48)

272


and

[2ia2(pi + 1) + qdan_i=(pi n+ 2) (pi n+ 1)an-2

a

E qkan_, for n=2, 3, . . (9-49)k=2

Equation (9-48) is the usual linear indicial equation. And substituting thisinto the recurrence relation (9-49) shows that

8iq03/2 (n 1) an -1= [qi+ 2iqi/2 (n [qi + 2iqo'/2 (n 1) [an

4q0 E qkan-kk=2;.

for n=2, 3, . . . (9-49a)

This recurrence relation will either terminate or else it will determine thean so that the series (9-47a) diverges.

In any event, the two formal solutions (9-47a) and (9-47b) will nowprovide, through equations (9-44) and (9-45), two normal solutions toequation (9-41), say w1 and w2, which are given by

00

W1 e ° z zPi E anz -71n=0

00

w2= E bnz-nn=0

(9-50)

Note that if condition (9-43) did not hold (i.e., if qo = 0) but the conditionq1 0 0 did hold, then equation (9-41) would be of rank 1/2 at z= co. In thiscase, it would not be possible to carry out the procedure just described.

273


9.6 SUBNORMAL SOLUTIONS

We have seen that if equation (9-31) is of rank r with r > 0 at the pointzo, it will have a formal solution of the form (9-38a) or (9-38b) only if r is aninteger. However, a change of independent variable of the type

4=ille

if zo is finite

if zo = co

where k is a positiv, integer, will usually transform an equation of fractionalrank into one ,whose rank is an integer. The formal solutions obtained by thisprocedure ;Ire called subnormal solutions.

Thus, we have shown that equation (9-41) in the preceding example hasrank 1/2 at co when condition (9-43) does not hold but the condition qi 0 0 doeshold. That is, the equation

where

and

w" q(z)w= 0 (9-51)

q(z) = E gizz- (9-52)n=1

q10 0 (9-53)

has rank 1/2 at z= co and, therefore, does not possess a normal solution at thispoint. However, upon making the change of variable = z'12, equation (9-51)becomes

12w _1 dwcr.42

+Q(4)w=o

274

(9-54)

where


Q(4) =4 E 4-2("-1) qn (9-55)n=1

And since this equation is of rank at V-- co, we introduce the change ofvariable

to obtain the equation

where

Then by putting

w= edict/ (9-56)

du42

d2u + Pi(4) Qi(4)u=0d

P (4) = 2co

colQi(0 w-1-4qi+ 4 E q4-2(n -1)n=2

(9-57)

aii+4q1=0 (9-58)

we find that the highest power of which occurs in Qi is less than the highestpower which occurs in Pi. And, as in the preceding'example, correspondingto the two roots ± i2q1I2 of equation (9-58) we obtain the following two equa-tions from equation (9-56):

d2u1 44_1121 dui I iq112(14 20 -0] u1 =0) L n

275

DIFFERENTIAL EQUATIONc

d2U2 ,+C .

d/L2 _ iq1121)] 0

dC2+4

j dCu2 =0+ q4 2

L 2 rfi2

But we know that these equations possess the formal solutions u1= CPI E anc-nn=0

and u2= CP2 E b4-", whose coefficients can be calculated recursively in then=0

usual way. And therefore equation (9-51) possesses the two subnormal solutions

w1= ei29li2z1J2z(P1 /2)Eanz n/2

n=0

tu. 2 c 1i2q112z1 I2 z(p2/2) -b.."z-n/2

n=0

9.7 NATURE OF RIGOROUS PROOFS

(9-59)

(9-60)

The formal solutions obtained in the preceding section are asymptoticexpansions of certain solutions to the differential equations. One methodfor actually proving this was developed by G. Birkhoff. In this method theleading terms of the partial sums of each formal solution are used to con-struct a homogeneous differential equation whose solution is known and whichin a certain sense is close to the given equation when z is near98 zo. This equationis then used to construct a singular integral equation whose solution also satis-fies the given differential equation. It is then proved that the solution to thisintegral equation possesses an asymptotic expansion which coincides withthe formal solution obtained by the methods discussed previously. For a de-tailed discussion of this method, in which equation (9-41) is treated to illustratethe general principles, the reader is referred to reference 32.

9.8 CONNECTION OF SOLUTIONS: STOKES PHENOtatfON

In the preceding examples, we have shown how to obtain the asymptoticexpansions as z co of two solutions to a differential equation of the type

98 Without loss of generality, the point zo (about which the solutions are obtained) is taken as co.

276


(`0 -41). The problem frequently arises of connecting these asymptotic expan-sions with an expression 99 for some given solution to the differential equationwhich is valid for small values of z. We know, in general, that the asymptoticexpansion of any given solution w to equation (9-41) can be expressed as alinear combination of the two asymptotic solutions (9-50). However, it usuallyturns out that the particular linear combination of these two asymptotic solu-tions used to represent the asymptotic expansion of the given solution w mustbe changed as the variable z crosses certain "critical rays" in going from onesector of the complex plane to another. This is the Stokes phenomenon which wehave mentioned previously.

For example, the change of variable

transforms Bessel's equation

into the equation

W--=z--1/2w (9-61)

z2W" + zir + (z2 p2) W = 0 (9-62)

1 2

P( 1 +4z

2 w =0 (9-63)

which has the form of equation (9-41) with the coefficients in equation (9-42)given by

and

qo = 1 q1 =0 1= p2

= 0 for n = 3, 4, 5, .

99 Such as a power series expansion.

(9-64)

277


And if we choose the arbitrary constants ao and be, respectively, to be

ao =2)1/2 1

-2 e-iRinr/2)+(04)]

2)2

'12 11)0 eiRP7112)+ (44)]

IT

it is easy to show that the formal solutions (9-50) become in this case

where

( 112 1 eiiz-(p.12)-(7,14)] E an( 0 n Z"7T 2 n=o

2)112 1w2= e-i[z-(p7d2)-(7rPoi t° anconz.(7r 2 n=0

an = (1-P) (1+ )2 n 2 P n

2nn!

(9-65)

(9-66)

(9-67)

Hence, the asymptotic expansion of any given solution W of equation(9-62) can be expressed as a linear combination

W(z) Ciz-h/2w1(z) + C2z-'12w2(z)

of the functions (9-66) and (9-67) multiplied by z-1/2. Let us choose the solutionW(z) to be the Bessel function of the first kind Jp(z). Then equation (8-37)shows that for 7r < arg z< 7r, C1= C2 = 1 and

(z) z ,112w (z) z 112 w2 (z) 7T < arg z < 7r (9-68)

This representation does not apply along the critical ray arg z = 7r, that is,along the negative real axis. To find an asymptotic expansion which is validin a region which includes the negative real axis, put

278


z= erizi (9-69)

Then since arg z= arg z1, arg z varies between 0 and 27r as arg zi variesbetween IT and 7r. And it follows from the definition (8-27) of the Besselfunction that

Jp(z) =Jp(effizi) = erPifp (zi) for jarg < 7r

But arg zi is in the range where Jp has the asymptotic representation (9-68).Hence,

Jp(z) eriPzi11/2wi (z ) + (71) for arg zil < 7r

And since equations (9-66) and (9-67) show that

w1 (Z1) = i[P" (742)62 (z)

it follows that

Jp(z) z-1/2 (z) e2iripz-1/2wi (z)

w2 (Z1) = eirinr+(1T12)11V1(z)

for 0 < arg z < 27r (9-70)

Comparing equations (9-68) and (9-70) shows that different linear com-binations of w1(z) and w2(z) must be usPri to represent the asymptotic expan-sion of Jp(z) in the sectors IT < arg ;. it and 0 < arg z < 27r. And, equa-tions (9-66), (9-67), and (9-70) show that

Jp(z) eirip+(37,ll2)(2

sin (z +2 4

to one term, for 0 < arg z < 27r (9-71)

Since in each region of the complex plane one of t unctions, w1(z) andw2(z), will be exponentially small compared with th er, the expansions(9-69) ai:d (9-70) will be equal to one another, with exponentially small error,in every region where they are both defined.

2 79Peo

CHAPTER 10

Expansions in Small and Large Parameters:

Singular Perturbations

Many of the differential equations encountered in practice contain param-eters. Although it frequently happens that we cannot find the exact solutionsto these equations, the behavior of these solutions for large or small values ofthe parameters is often physically important. In such cases we are content tofind asymptotic representations of the solutions which are valid for theselimiting values of the parameters. A number of techniques for obtaining suchexpansions are given in this chapter. Since no formal theory has been developedfor many of these techniques, the approach of this chapter is necessarilyheuristic and the material is frequently presented by means of examples.On the other hand, the ideas developed herein are quite general; and theyapply not only to ordinary differential equations, but also to partial differentialequations, integral equations, and even difference equations.

10.1 NONSINGULAR EXPANSIONS OF SOLUTIONS

The general second-order homogeneous linear equation containing alarge parameter X is of the form

d2y + p(x, X)dxdy

+ q(x,dx2

(10-1)

We suppose that p(x, X) and q(x, X) have (at least formal) power series ex-pansions in X. if these expansions contain only nonpositive powers of X, say

281


p(x, A) = Pn(X)Xnn=0

q(x, X) = qn(x)X-nn=0

then the solutions to equation (10-1) will have a formal power series expansionof the form

Y(x)= yn(x)A-nA=0

(10-4)

where the functions yn(x) are determined by substituting the expansion(10-4) into equation (10-1) and equating to zero the coefficients of like powersof A to obtain

n-1 f \ dyk ,dynPo(x) qo(x)yn=- [Pn-kkX) (in kkXilYk]dx- k--=0 dx

for n =0, 1, 2, . . . (10-5)

and the sum on the right is omitted for n=0. These equations can, at least inprinciple, be solved successively for the coefficients yn. If the series (10-2)and (10-3) converge, it can be shown (ref 4, p. 126, ex. 6) that the series (10-4)will also converge. In any case, the formal series (10-4) will usually be anasymptotic expansion.

However, if the formal power series expansion for either p(x, A) or q(x, A)involves any positive powers of A, the expansion of the solution in powers ofA will, in general, involve infinitely many positive and negative powers of A;and it will not be possible to solve for the coefficients successively. Neverthe-less, it is still possible in certain instances to obtain formal solutions whoseterms can be calculated successively by using a technique analogous to theprocedure used in the preceding chapter to obtain solutions at irregular singularpoints. These solutions will not, in general, be convergent series; but theywill be asymptotic expansions. Thus, consider the equation

d2y dydx2+ p(x, X) d7+ q(x, h)y=0 (10-6)

282

where

xp(x, A)= po(x)Xk- n

n=0

xq(x, x)= E (x),k2k nn=0

SINGULAR PERTURBATIONS

(10-7)

(10-8)

k is a positive integer, and po(x) and q0(x) are not both identically zero. Thenit can be shown by ;Iirect substitution that equation (10-6) will possess twoformal solutions of the form

where

en(x) yri(x)X-n (10-9)/I = 0

k -1fl(x) = con(x)Xk-nn-7.0

(10-10)

The coefficents yo and oh can be calculated by solving successively the setof ordinary differential equations which is obtained by substituting the assumedsolution (10-9) into equation (10-6) and equating to zero the coefficients oflike powers of A.

In order to illustrate these ideas, consider the differential equation

d2ydx2

+[A240(x) + q2(x)] y= 0

which was first discussed by Liouville in his classical investigations of theSturm-Liouville problems.

Since in this case k = 1, the differential equation will possess formalsolutions of the form

xy= ewo's)A yn(x)X-n (10-12)

n=0

283

488-942 0 - 73 - 19


After substituting this into equation (10-11) and collecting terms, we find that

E (A+ 42y.),\ n+ E`a (2coVi + (OP)rn)X1-71 Ec° (qo + 0.42)y,,X2-n = 0n=0 n=0 n=0

But upon shifting indices, this becomes

(Y11-2+ q2Yn-2)A2n + (264'n_1 + cOPYn-1)n=2 n=1

2n

+ E (go + wto2)y.X2-" = 0n=0

And by equating the coefficients of the like powers of A to zero, we get

(qo + 042)yo = 0 for n = 0 (10-13)

2(.4y; + c6gYo + (qo + cu 2)y1 = 0 for n = 1 (10-14)

and

Y;;--2 42Yn-2 2coLYn_1 + cogyn-i (qo + 42)yn = 0 for n = 2, 3, . . .

(10-15)

In order to avoid the trivial solution y=0, we must require that yo 00.Hence, equation (10-13) becomes

qo + c42= 0 (10-16)

When this is used in equations (10-14) and (10-15), we obtain

2644+ 4yu= 0

adLy;,_, cogYn--1 = Ynti2 q2Yn-2 for n = 2, 3, . . .

284

10-17)


In the case of the differential equation (10-1) we had to solve the set ofsecond-order differential equations (10-5) to determine the coefficients yn;whereas, in this case it is only necessary to solve a set of first-order equations.'°°Since qo 0 equation (10-16) will have two distinct solutions, say coo,' andwo,2, given by

(do, f = + J -\rrio dx (10-18)

WO, 2 = "V740 dx (10-19)

Corresponding to each of these roots, we will obtain a different set ofequations from equations (10-17) and, therefore, equation (10-12) will yieldtwo different formal solutions to equation (10-11). It is proved in reference32 (p. 83), that these two formal solutions are actually asymptotic expansionsof two linearly independent solutions to equation (10-11) in any finite intervala < x < b in which qo does not take on the value zero. The first equation(10-17) will yield the same equation for both of the roots (10-18) and (10-19).Thus, when equation (10-16) is substituted into the first equation (10-17),we obtain the separable equation

y° + (-914qo°

)y =-0

which is easily solved to obtain

yo = constant X %T IM, (10-20)

Equations (10-18) to (10-20) can now De substituted into equation (10-12)to show that the general solution of equation (10-11) in any interval a < x < bwhere qo(x) 0 0 has the asymptotic expansion

! v=y CLQ`I'4e q.

di -4- C2q6 1/ 4e-4 V--7"11 to one term (10-21)

100 This situation is analogous to the reduction of the indicial equation from a quadratic to a linear equation ingoing from a regular singular point to an irregular singular point.

285


where C1 and C2 are arbitrary constants. Thus, qo(x) must be either strictlypositive or else strictly negative for all a < x < b. If qo is strictly negative, theconstants of equation (10-21) can be redefined slightly to obtain

Y C3 (go) 1/4eAfv:---q.dx + C4 ( qo)-1/4e-k V7.6

to one term, for qo(x) < 0 (10-22)

And if qo(x) is strictly positive for all a < x < b, the constants can be redefinedto obtain

y Ciqi.-1/4 sin (x f Vq;dx)+ C2q;;'I4 cos (1. J NiTriodx)

to one term, for Q0 (x) > 0 (10-23)

Notice that the asymptotic solution (10-22) has a monotonic behavior,while the asymptotic solution (10-23) has an oscillatory behavior.

10.2 TRANSITION POINTS

Now let us consider the case where qo(x) is equal to zero at a single point,say xo, in the interval a < x < b. And suppose, in addition, that qo(x) andq2(x) are analytic at xo. Then qo(x) and q2(x) must have the expulsions

qo(x) = qo,i(x xo) qo,2(x x0)2 + . . (10-24)

q2(x) = q2,o+ q2,1 (X X0) + (10-25)

about the point x= xo, and we shall require for definiteness that qo,i 0.Thus, the point xo is an ordinary point of the differential equation (10-11).

Every solution to this equation must, therefore, be analytic at xo. But in viewof equation (10-24), equation (10-21) implies that

0 ((x zo)-114) as x ---> xo (10-26)

Hence, the asymptotic expansion (10-21) cannot represent an analytic func-tion in a neighborhood of xo and, therefore, it certainly cannot be the asymptotic expansion of any solution to equation (10-11) in the neighborhoo?this point.

286


However, for any positive number 6 (no matter how small), equation(10-21) provides a valid asymptotic expansion, to one term of the generalsolution .o equation (10-11) both in the interval a < x < xo 8 and hi theinterval xo + 8 < x < b because qo(x) does not vanish in either of theseintervals. Since ecuation (10-24) shows that qo(x) is not tangent to thex-axis at xo, it must cross the axis at this point and must, therefore, bze positivein one of the intervals. The solution to equation (10-11) will then have theasymptotic expansion (10-23) in this interval. In the other interval, qo(x) willbe negative and equation (10-11) will have the asymptotic solution (10-22).Thus, the asymptotic solution to equation (10-11) will have an oscillatorybehavior on one side of xo and will have a monotonic behavior on the other side.The transition from one type of behavior to the other takes place in a smallregion centered at xo. The point xo is, therefore, called a transition point.'"

When the second equation (10-17) with n =2 is solved for yi(x) andequations (10-16) and (10-20) are substituted into the result, it is foundfrom equations (10-24) and (10-25) that

yi=0((xxo)-7/4) as x --> xo

But equation (10-26) shows that yo is of order (x x0) -"4 as x > xo. Hence,the second term in the expansion (10-12) will be approximately equal to thefirst term when

I (X x0) 1!4I X-11 (X X0)-714 I

that is, when the distance between x and xo is approximately Ix xol = X-213-Thus, when the distance between x and xo is less than or .,qual to X-2I3, the

order of the terms of the expansion (10-12) will actually increase with increas-ing n. This shows that the asymptotic expansion breaks down in a region ofradius X-2/3 about the point xo. It is, therefore, said to be a nonuniformly validasymptotic expansion.

In order to obtain an asymptotic expansion which is valid in this region, werescale the independent variable so that it will be of order 1 therein. That is,we introduce the new independent variable

6= (x xo) X2!3 (10-27)

"The term turning point is also used. This term arises from the application of eq. (15-11) ,o classical and quan-turn mechanical wave reflection pioblems.

287


into equations (10-11), (10-24), and (10-25) to get

and

C121+ [ x2/3ande -a ) A4/3 92 ( A213)] y= 0

z40 ( x213 )= 40,1 x213+ go, 2 x213 ±

X213 A2/3 (10-30)

Then upon substituting equations (10-29) and (10-30) into equation (10-28)and neglecting terms which are small for large values of A, we get

2d+ go, ley= 0 (10-31)

The solution to this equation should be "close" to the true solution to equation(10-11) at least in the region

< X-2/3 (10-32)

or 14) < 1. Hence, it should represent the first term of the asymptotic expan-sion of the solution to equation (10-11) in the region (10-32).

The differential equation (10-31) is a form of Airy's equation (ref. 29,section 6.4) which can be transformed by the change of variable

into the Bessel's equation

288

4=3 V407 e3/2


odd2:72+ ado_ [42 _ )2 I Y= o

(see eq. (8-24)). Thus, the general solution to equation (10-31) is

1/2

3y= X113 D1e2j1/3 2 63/2 )

2+ x1/3 )112 D2e1/2./-1/3 ( -3 e3/ 2 ) (10-33)

where D1 and D2 are arbitrary constants and the normalization factor (iTX1/3/3)1/2has been inserted for convenience.

Let us now suppose, for definiteness, that

go, 1> 0 (10-34)

Then solution (10-22) will hold in the region a < x < xo 8 and the solution(10-23) will hold in the region xo + S < x < b, where 8 is some arbitrarilysmall positive constant. The solution (10 -33) holds in some region which iscentered at xo and has a size at least of order X-2/3. Having found the piecesof the solution which apply in the three regions into which the interval a < x < bhas been divided, it is now necessary to match up these pieces across the ad-jacent regions to form one continuous solution. These pieces of the \solutioncontain altogether six arbitrary constants. Four of these will be determined bythe matching requirement. The other two must remain arbitrary if we areto obtain an asymptotic expansion of the general solution to equation (10-11).In order to accomplish this matching, we suppose that the regions of validityof the various pieces of the solution can be extended in such a way that anytwo adjacent regions overlap one another. Thus, we assume there is an "over-lap domain" (or intermediate region) in which both the expansions (10-22)and (10-33) are asymptotic expansions of the solution to equation (10 -11)and that there is an overlap domain in which both the expansions (10-23) and(10-33) are asymptotic expansions of the solutions to equation (10-11) (see

289


Region of applicability of solution 110-33), "inner region"

"Outer region", region of ap- "Outer region", region of appli-plicability of solution 110-221 cability of solution 110-231

;\\-\\\\ al/r10000(\\\\\\WbOverlap Overlapdomain domain

FIGURE 10-1. Regions for transition-point expansion.

fig. 10-1). We therefore require that within the overlap domains the twoadjacent expansions agree with each other to within an error which is ofsmaller order than the last term retained in these expansions.

Since the size of the region in which the solution (10-33) applies (whichwe will call the inner region) approaches zero as X--> 00, the location of thetwo overlap regions must also approach xo as A - 00. Now in the outer regions,the asymptotic expansions correspond to holding the variable x fixed andtaking the limit A - 00. In the inner region, the asymptotic expansions corre-spond to holding the variable e= (x xo)A213 fixed and taking the limit A -*co.This limiting process allows the coordinate x in the inner region to move towardxo as the size of this region shrinks to zero. Thus, the matching can be per-formed in a precise manner by introducing into both the inner and outer ex-pansions an "intermediate variable"

= X0) X0 = eXEct-1213)] (10-35)

with a chosen so that a fixed value of 71 will remain in the overlap region asthis region shrinks toward xo with increasing X. The matching of the adjacentexpansions is then accomplished by changing their independent variablesfrom x and e to 71 and then requiring that these expansions become identical(to the appropriate order in X-1) when the limit A- 0o is taken while holding71 fixed. However, if 71 is to lie in the overlap regions, we must require that

0<a<-3

290

(10-36)

SItt;GULAR PERTURBATIONS

Now in order to apply the matching procedure to this problem, we intro-duce the new variable (10-35) into equation (10-33) to obtain

y=(3 "1"a ) 1/2

Doi 1/2 j.,3

t_ 2,7, 7)3/2X[(2/3)- a13 /2}

""

7r 2+ (-5' XI-. D27)1/2J-1/31-VWi 713/2x [(2/3)-(43/21

3 '(10-37)

First, suppose that 7) > 0. Then equation (8-37) shows that in the limit X --> ccwith 7) fixed

Xa 1/477-

q0,171 ) D1 e" 2V4Ti'l73/2X[(2/3)-a13/2 }5

3 12

Xa± D2 COS { Nr- "07312X,(2 3) ai3/ 2 73.12} to one term

(10-38)

But for 71 < 0 we find that 773/2=ei(37/2)17)13/2; and therefore the argument ofthe Bessel's function in equation (10-37) is no longer in the range where equa-tion (8-37) holds. We must, therefore, use equation (9-71).

In this case, we see that in the limit as X oo with , fixed

1 )1/4 2

(D2D1) exp N/1.1071" n 13/2 x[2/3,43/21

Xa )1/4I X

1

13/2 12/3-a]3/21D e -i716 + D2 ei1116 exp 2 V2 q0,1711

to one term (10-39)

291


Next substituting equation (10-35) into equation (10-23) and neglectingthe higher order terms in X-1, we find that

xac

1/4

V lqcon . sint2

3 TC1Z1 713/2 XI(213)cd3/2

Xa 2

3+

Q0C2 C2 cos {- Wo-; 7)3/2 XU2/3)-a13/21 (10-40)

Hence, the expansion (10-38) and the expansion (10-40) will become identicalin the right overlap domain if

57r . irC1= DI sin 12 + D2 sin

12

57r 7rDi cos 12 + D2 COS 12

and the expansion (10-23) becomes

y Die/V/4 cos (1l is V dx + D2 qT114 cos ( X1 dxso 12 xo 12

to one term, for xo < x < b (10-41)

In order to match the solutions in the left overlap domain, we must considertwo cases. First, suppose that D1 D2. Then the second term in equation(10-39) is negligibly small compared with the first and may be neglected.Hence, equation (10-39) becomes

1 Ka \ 1/4

2 q0,717) I ) ( D2 D1) exp 3 q0,117)13/2X[(2/3)--a]3/2

(10-42)

This must now match with equation (10-22) in the left overlap domain. Itcan be seen that this matching can occur only if C3 0. In this case, thesecond term will be exponentially small, compared with the first; and equation

292


(10-22) becomes, upon substituting in equation (10-35) and neglectinghigher order terms in 1/A while holding fixed,

y C3( Xa I1/4 exp 2 V I n 13/2 XR2/3)a13/2

q0,1171Igoo

But this will match with equation (10-42) in the left overlap domain only ifC3 = (D2 D1) /2. And the solution (10-22) therefore becomes

D2 D1r ~

2( q0) -1/4 exp o dx

xoto one term,

for a < x < xo and Di 0 D2 (10-43)

Next consider the case where D1= D2. Then the first term in equation(10 -39) is zero and, therefore, the second term cannot be neglected. Equation(10-22) will only match onto equation (10-39) in the left overlap domainif C3 = 0 and C4 = (1C3/2) D1. Therefore, equation (10-22) becomes

Y V-3- Di (- (1°)-114 exP (- X Is

dx)2 xo

to one term,

for a < x < xo and D1= D2 (10-44)

Thus, the complete asymptotic expansion of the general solution toequation (10-11) in the interval a < x < b iP given by equations (10-33),(10-41), and (10-43) or (10-44). The expansions (10-41) and (10-43) or(10-44) become poorer and poorer representations of the true solution as

xo (they are nonuniformly valid expansions), but the expansion (10-33)provides a good representation in this region. It is possible to obtain a uniformlyvalid asymptotic expansion in the region xo x < b by adding the expansion(10-41) to the expansion (10-33) and (so that it will not be -minted twice)subtracting their common value in the transition region given o. equation(10-38). Thus,'°2

Yuniformly valid eq. (10-41)+ eq. (10-33)- eq. (10-38)

102 By construction, eqs. (10-33) and (10-38) are asymptotically equal in the outer region.

293


Similar remarks, of course, apply for the interval a < x -.-. xo. Although thismethod is very general and can be used even when a larger number of terms areretained in the asymptotic expansion, there is a better method of obtaining auniformly valid one-term asymptotic expansion for the present problem which,in fact, applies to the entire interval a < x < b. To this end, notice that the

fX

s Wterm A dx becomes asymptotically equal to the term0

2A

2V q0, 1 (X XP) 3/2 .=--g Vq0, 63/2

in the transition region. It is, therefore, reasonable to hope that the range ofvalidity of equation (10-33) (which applies for small values of 103 x- xo) can beextended by replacing 2/3 17q0,1 e3/2 in the arguments of the Bessel functions by

JXf Nric, dx and replacing the factor X1/6 V-e-by,q0-114 viodx)1/2.uponxo B so

making these substitutions, equation (10-33) becomes

1/21/4 ( .fj- X Arq; dx 11 /z [DJ1/3 (X IX VQ; dx) + D2.1-0 (X fx Nri0 dx)]" 2 xo xo xo

(10-45)

Having obtained this result by a heuristic argument, it is now easy to verifythat, for X2/3(x xo) =0(1), this equation agrees with equation (10-33) to thelowest order A-1 and, for x- xo= 0(1) with x > xo, it agrees to lowest order inA-1 with equation (10-41). Similarly, when x-xo=0(1) and x < xo, it agreesto lowest order in A-1 with either equation (10-43) or equation (10-45), de-pending on whether D1 is equal to D2. Thus, equation (10-45) must representa uniformly valid asymptotic expansion to one term of the general solution toequation (10 -11). This result was first obtained by Langer (refs. 33 to 35) by amore formal procedure. The technique of using inner and outer expansions toobtain a uniformly valid expansion began with Friedrichs (ref. 36) in the 1950'sand was developed by Kaplan, Lagerstrom, Cole, and many others. Thistechnique not only applies to linear and nonlinear ordinary differential equa-tions but also to partial differential equations. Many of the applications of thismethod have been to fluid mechanics problems. In fact, the basic ideas grewout of boundary layer theory.

1°3 This is because is of order 1 and, therefore, for large X, x must be close to xo.

294


10.3 MATCHED ASYMPTOTIC EXPANSIONS

The method of using inner and outer expansions to obtain a uniformlyvalid expansion can, in fact, be applied to obtain solutions in a systematicway to certain equations which are of the form (10-6). The solution (10-9)to equation (10-6) was obtained by essentially guessing its general form.Therefore, this procedure is limited to equations of the type (10-6); the methodof matched asymptotic expansions has no such limitation.

The method is usually applied in such a way that it is necessary to considerthe boundary conditions along with the differential equation. The ideas in-volved are best illustrated by means of an example. However, instead of con-sidering an equation containing a large parameter we will now consider anequation containing a small parameter 104 E. Thus, we shall seek an asymptoticexpansion as E 0 of the solution to the equation

dyE

d2y w+ q(x)y= 0

subject to the boundary conditions

(10-46)

y(0) =1 (10-47)

y(1) = a (10-48)

and where the functions p(x) and q(x) are any functions which can be repre-sented by power series, say

P(x) pnxn q(x) E qux" (10-49)ra=0 n=0

near x = 0, and by similar power series, say

p (x) = x)"- q(x) = E qn(1x)n11=0

104 Of course, since we can always put X=1/e to obtain an equation containing a large parameter, there is noreal difference involved.

295


near x= 1. We shall also require that

p(x) > 0 for 1 (10-50)

and that the integral 11 [q(x)/p(x)]dx exists..0

Since E is small, it is natural to seek a solution to equation (10-46) in theform of a power series in E. Thus, let us try to obtain a formal solution to equa-tion (10-46) of the form

Y(x; E) = Y (10-51)n=0

Upon substituting this into equations (10-46) to (10-48) and equating to zerothe coefficients of like powers of E, we find that

dyop(x) + q(x)yo= 0dx

P(x)(x

d2Yn -1) dx

ALIq(x)Yn dx2

yo(0) = 1

for n= 1, 2, .

(10-52)

. (10-53)

(10-54)

yo(1)=a (10-55)

Yn(0)=yo(1)=--- 0 for n=1,2, . . (10-56)

Since equations (10-52) and (10-53) are first-order linear equations, theycan be solved immediately to obtain

Yo(x) = Coe-n(x)

[ fxp(

1 1dx) dx2 x j

296

(10-57)

fo'rn =1,2, . .

(10-58)


where the Cn for n= 0, 1, 2, . . . are arbitrary constants and we have put

SI(x) fx dxP(x)

(10-59)

Notice that since the integral S2(0) exists by hypothesis, the integral (10-59)must certainly be finite for all x in the interval 0 x 1.

Now, except in the very exceptional circumstance where a = eno), it isimpossible to choose Co so that the zeroth-order solution (10-57) satisfies bothboundary conditions. Thus, suppose that a 0 64" and, therefore, that one ofthe boundary conditions (10-54) or (10-55) cannot be satisfied. We reason that,just as the expansion (10-12) broke down at the transition point, the expan-sion (10-51) will break down at one of the boundary points x= 0 or x=1and it will be necessary to obtain a different (inner) expansion in this region.105This expansion must then be matched smoothly onto the outer expansion(10-51) in some intermediate region. It is necessary to investigate the be-havior of the asymptotic solutions at both boundaries in order to find whichone will correspond to the boundary layer region. If this is done, it will befound that, due to the condition (10-50), it will be impossible to match anyinner solution which occurs near the boundary x =1 to the solutions (10-57)and (10-58). This is due to the fact that condition (10-50) will cause all possiblesolutions for the region near x=1 to grow exponentially with the distance 1xand this type of behavior is ir.compatible with the solutions (10-57) and (10-58).

The constants Cn for n = 0, 1, 2, . . . must, therefore, be determinedso that the boundary condition (10-55) and the second boundary condition(10-56) at x =1 are satisfied. Hence, Co= a and Cn= 0 for n= 1, 2, . . .; andthe solutions (10-57) and (10-58) become, respectively,

yo(x)=ae-ox) (10-60)

Y n(x) e-a(x) j" sen(x)Pfix) dx2

d2Yni dx for n= 1, 2, . . . (10-61)

The functions yn(x) for n= 1, 2, . . . can now be determined successively bysubstituting the expression for obtained in the previous step into the rightside of equation (10-61). Thus, substituting equation (10-60) into equation(10-61) with n =1 gives

05 Which is frequently referred to as the boundary layer region.

297


Yi(x) ae-ncr) 12. 1

1 p (x

where we have put

r(x) =q(x)p(x)

and we shall suppose that the integral

Aqp(x) [r2(x)- r' (x)]clx

(10-62)

(10-63)

(10-54)

exists.Unlike the situation which occurred at the transition point, the zeroth-

order solution remains bounded at x =0. In fact, its Printing value is ae-flo).But in order that the boundary condition (10-47) be satisfied, it is necessarythat the asymptotic solution change from the , glue 1 to the value ae-n(°) acrossthe boundary layer region. Now we antic' ,ate that the expansion (10-51) Willhold over most of the region 0 x 1 but that it will break down in a narrowregion near x= 0 whose thickness approaches zero as 0. Since the asymp-totic solution y must change from ae-no) to 1 across this very narrow region,the derivatives y' and y" must become very steep (large) in this region. How-ever, in assuming that the solution had an asymptotic expansion of the form(10-51), we were essentially treating the terms )/', py' , and qy in equation(10-46) as if they were of order 1. In order to overcome this difficulty, we pro-ceed just as in the case of the turning point and rescale equation (10-46) in amanner which is appropriate to this boundary region by introducing a new"stretched" independent variable x by

x(1)(E)

(10-65)

where urn 0 and the function 4 is to be chosen so that it is of the scaleof the boundary layer region. In this case, however, it is necessary to determinethe scaling function 4 in a different manner than in the case of the turning point.Thus, upon substituting equation (10-65) into equations (10-46) and (10-49),we obtain

298


E Cl!,F+ p(40(E)21412 (E) 4,(E) dg q41()i)i, = 0 (10-66)

where we have put 'Ai; E) = y and, in view of equations (10-49) and (10-50),

P(4)(e)R)= Po + P14)() + P2432(e)R2 + . .

for po 0 0q(4)(E)x)= go + qict)(E)g + q24)2(e)22 + . . .

(10-67)

Now in order that the inner solution satisfy the boundary condition atx= 0 and still contain another constant which can be adjusted to match theouter solution, we must require that the equation which is satisfied by thelowest order term in the expansion of the solution in the inner region (the innerexpansion) be of second order. This equation is obtained by holding x and yfixed and taking the limit'06 0 in equation (10-66). However, the limitingform of this equation depends on the choice of the function OW. And in orderthat the highest order derivative be retained in this equation we must requirethat

0=0(0 (10-68)

If the condition 41= o (E) also held, the second and third terms of thelimiting equation (10-66) would drop out and we would be left with the equation

However, if we require that

E 4.21_4)2(E)

(10-69)

00o(E) (10-70)

then both the first and second terms will be retained in the limiting form ofequation (10-66). In addition, the size of the inner region (which is determinedby 4)) will be larger if condition (10-70) holds than if it did not hold. Thus, the

"6 This limiting process is called the inner limit. The limit taken while holding x and y fixed and letting 0 iscalled the outer limit.

299

488-942 0 - 73 - 20


solution which would be obtained by assuming that condition (10-70) did nothold is a limiting case (for small i) of the inner solution which is obtained whenequation (10-70) does hold. We shall, therefore, assume that conditions (10-68)and (10-70) hold. And since only the order of magnitude of 4 is important, wewill lose no generality by putting 4)() =E. Hence, equation (10-66) becomes,upon inserting equation (10-67),

2f0 (10-71)(po epii+ 2p212 + . . .) + (Ego e2q11 . . )y =

We now suppose that the solution to equation (10-71) has an asymptoticexpansion of the form 107

00

5(x; ) E 5.rn Wenn=0

(10-72)

Then upon substituting this into equation (10-71) and equating to zero thecoefficients of like powers of E, we obtain

d2yo_L dyod12 p0 di=u

d2yn dyna2 -P° cig="71M for n= 1, 2,

where we have put

XPnk+ Ykqnki) Xnk-1177,(1)_V ( dykaxk=0

for n= 1, 2, . .

(10-73)

(10-74)

(10-75)

And upon substituting equation (10-72) into the boundary condition (10-47),we find that

yo(0) =1 (10-76)

107 It should not be concluded from the results obtained so far that the asymptotic expansions of the solutions toequations containing a small parameter will always be power series in the parameter. For example, logarithmic termscould occur.

300


5 (0) = 0 for n=1, 2, . . (10-77)

Since equations (10-73) and (10-74) do not contain the dependent variableexplicitly, they can easily be integrated by the methods of chapter 4 to obtain

yo= Eo boe-P°'

Yn= En+ -6ne-P-1o 0

+ f [1- P°(5-2)]17P

n(y)dy for n=1 2

which become, after applying the boundary conditions (10-76) and (10-77),

Yo eo+ (1 E0)CP°' (10-78)

= (1 e't) +1 [1 eP°() (57)0 for n= 1, 2, . . .po

(10- 79)

We can again determine the functions yn for n= 1, 2, . . . successively bysubstituting the expressions for yn_i obtained in the previous step into the rightside of equation (10-79). Thus, upon substituting equation (10-78) into theright side of equation (10-79) with n= 1 and carrying out the integration, weobtain

(1 co)jr-1 = CI (1 e-PoR) ieP°5 14o pi

PO

where we have put

pipo qo-) Eo o2

ci =ir

+ 7, ( 1 co) (Pi q0) + Eoqo]

(10-80)

301


As in the case of the transition point, we now suppose that there is anoverlap domain in which the outer expansion (10-51) and the inner expansion(10-72) both apply and that within this overlap domain, these two expansionsagree to within an error which is of smaller order than the last term retained.In order to perform this matching, we introduce an intermediate variable

ExX+ =

x=.-

v(E) (E)(10-81)

into both the inner and outer expansions. The scale factor v(e) is determinedso that the variable x+ is of order 1 in the overlap domain. Since the size ofthe overlap domain must approach zero as E 0 and since it must be fartheraway from x= 0 than the inner regions whose size is 0(e), we must require that

and

E= 0 (V) (10-82)

v=o(1) (10-83)

To proceed with the matching, we then reexpand the first m terms for m=1, 2,. . . of both the inner and outer expansions while holding x+ fixed and thenrequire that the difference between these two expansions be of o(em-i). Thatis, the inner and outer expansions are asymptotically equal in the overlapdomain to the appropriate order. Hence, we require that

yo[v(e)x+] + eyirv(e)x+] + . . . + enyn[v(e)x+] Yo [40x+i

Eyl v(Ee) = 0 (En) as E > 0,re)

[ .X+] . . . EnYn XI]

for n = 0, 1, 2, . . . with x+ fixed (10-84)

Thus, for example, when n= 0, this becomes

Yo[v(E)x+] [1L(--1 x+] = o(1) as E 0 with x+ fixed (10-85)

302


And when n = 1, it becomes

Yo [v(E)x+ + Eyl [v(E)x+ [12-LC-1 x+] EY1 [ V ( =0(E)

Now since

a[v(E)x+] =

as E 0 with x+ fixed (10-86)

q(vx+) v dx+ = IPS+ q() cle = f° g-1p(vx+) 1 P(e) P(e)

vx+P(e)Isfl de= MO) + p(E)x+ +0(v2)

it follows, upon expanding the exponential, that

[e-fl[v(ox+i= e-flme-vx+00P0+ 0(v2)= e-110) 1 v(E) --c) x+ +0 (/2)]Po

And, the solutions (10-60) and (10-62) therefore become, respectively,

Yo[v(E)x+] = ae-non [1 v(E)go x+ +0 (v2)] (10-87)

yl [v(E)x+] = ae-11(0[4 +0 (v)] (10-88)

where definition (10-64) has been used. On the other hand, the solution (10-78)becomes

5P-0[1-ix+] = Eo+ (1 eo)e-Pdv(c)/(ix+

But, since equation (10-82) shows that

303


this becomes

v(E) = 00E

yo [ = 'do+ exponentially small terms

Similarly, the solution (10-80) becomes

YIPLE) x+an v (E)

x + exponentially small terms= el C° PO c

Equations (10-87) and (10-89) now show that

yo[v(E)x+ Yo[v(E)

X+ = ae-n" co +0 (V)

(10-89)

(10-90)

since the exponentially small terms are of lower order than any power of v. Butsince lira v(E)-= 0, this shows that the zeroth-order matching condition (10 85)will be satisfied provided that

eo = ae-no) (10-91)

Similarly, equations (10-86) to (10-90) show upon substituting in equation(10-91) that

yo[v(E)x+] + Eyi [v(E)x+] [v(E)

x+] [ x+]

ae (1 vqi

x+ + E.A) ae- n(o) (1 v qo x+) Eti+ 0 (v2)

e[e-n(o)aA + 0 (v2)

304


Hence, the first-order matching condition (10-86) will now be satisfied if we putae-M0A and choose the scaling function v so that

1,2= () (10-92)

Notice that v must simultaneously satisfy conditions (10-82), (10-83), and(10-92). There are many choices of v which will accomplish this. For example,we can take v(E) = 2/3.

We have now obtained the following results: In the outer region, the expan-sion to two terms is

y(x; E) GE-11(x) E 1(x)

[r2 (x) ri(x)]dx+ . . .1 (10-93)

In the inner region, the expansion to two terms is

Ex5r(g; E) e-Pc.'1[1- ae-n(0)][1+ (goPi P1P2(1t)] ae-f"ell

po

+ ae-a(0) (1+ EA RE2)+ . . (10-94)

And in the overlap domain, both of these expansions take on the commonvalue ye given by

yr - ae-a")) (1+ EA -Po x) (10-95)

As in the case of the transition point, we can obtain an expansion whichis valid everywhere in the interval 0 x -.5. 1 (that is, a uniformly valid expan-sion) by adding the inner expansion to the outer expansion and, so that it willnot be counted twice, subtracting their common value in the overlap domain.Thus,

1

)Yuniformly valid ae-n(x) 1 [ r 2 r' ( x )p(x

pip°2

ae-nco) 0. (1-96)e-Poi[1- ae-flm][1+ wo-Ei (go pi

305


Of course, we have not actually proved that equation (10-96) is indeed anasymptotic expansion of the solution to equation (10-46) subject to the boundaryconditions (10-47) and (10-48). However, it is common practice in appliedmathematics to accept an expansion obtained by a formal procedure (such asthose given in this section) as being a true asymptotic expansion of the solu-tion to the problem. Indeed, most problems which are treated in practice aretoo complicated for rigorous proofs to be carried through.

We emphasize again that this method applies to nonlinear equations justas well as to linear equations.

10.4 METHOD OF STRAINED COORDINATES

We shall now consider some additional techniques which can be used tohandle the nonuniformities that arise when asymptotic solutions are soughtto certain types of differential equations. Thus, consider the differentialequation

d2ydt 2

+ co2y = Ey3 (10-97)

which describes the oscillation of a mass on a spring with a weakly nonlinearrestoring force. And for definiteness, let us impose the initial conditions

6-11-y(0) == 1dt

(0) = 0 (10-98)

If .,ve attempt to find an asymptotic solution in the form of, a straight-foi wan power series in E

Y(t; E) yo(t) + Eyi(t) + . . .

then we get, upon substituting this into equations (10-97) and (10-98) andequating to zero the coefficients of like powers of E,

dd2Yt2

0+ w2y0 = 0

306

(10-99)

dt2

d2yi.+ co Y1 YO

yo(0) = 1dyo (0) = 0dt

yyi(0) = d(0) = 0

dti


(10-100)

(10-101)

(10-102)

However, the solution to equation (10-99) subject to the initial conditions(10-101) is yo = cos cot. And after substituting this into equation (10-100), weobtain

3d2y2 i+ co2y, =-- cos3 cot = 4 cos cot

4t + cos 3cot

d

But since the general solution to this equation is

y1- 1 tcos 3cot + -e

37 sin cot + ci cos cot + c2 sin cot

co

the asymptotic solution to equation (10-97) is

1y cos cot + e [

3 tsin cot

32co2cos Scat + CI cos cot + C2 sin cot + .

8 co

However, the first-order term in this expansion will not be small comparedwith the zeroth-order term when t 1/E due to the presence of the termt sin cot in the solution. Such terms are called secular terms. The expansionwill, therefore, not be valid for large times even though it may be suitable forcalculating the solution at small times. The solution of equation (10-97) isperiodic. However, due to the appearance of secular terms, the solution cannotbe carried to sufficiently long times to calculate the distortion of the perioddue to the nonlinear restoring force.

A method (based on the work of Poincarem) for alleviating this difficulty

The ideas were developed in the course of his work on periodic orbits of the planets.

307


was developed by Whitham and by Lighthill (ref. 37). The method dependsupon introducing a new variable T and then considering both the dependentvariable y and the independent variable t to be functions of T. Thus,

t(r; E)

y = y(r; E)

And we suppose that these functions can be expanded in powers of E to obtain

T± Eli (T) . . . (10-103)

y = yo(T) + Eyi(r)+ . . . (10-104)

The arbitrariness introduced into the solution by the functions ti, etc., is usedto adjust the nonhomogeneous terms in the equations for yl, y2, . . . so thatthe secular terms will not occur in the expansion. The resulting distortion ofthe time scale will cause the frequency of the motion to depend on E. And thiswill allow us to calculate the variation of the period.

Now it follows from equation (10-103) that

.dt dt dr (1 + . . .) dr Et;+ . .) (10-105)

where the prime denotes differentiation with respect to T. Hence,.

dd2T22(1 Et; . . .) ar [(IEt; . .)

k dr d.)]

= (1 2Et; + . . )h2d7-2 E dir2 Et; ddY: . . .

After substituting this into equation (10-97) together with the expansion(10-104) and equating to zero the coefficients of like powers of E, we obtain

d2Yo

dr2

308

(10-106)

SINGULAR PERIURBATIONS

d2yi d2Yo dyo+ oPyi = y3 + 2t1 --+ drdr2 ° dr2(10-107)

In order to find the proper boundary conditions, we need expressions forthe values of y and dyldt at t = 0 in terms of their values at T = 0. Thus, we usea Taylor series expansion of these quantities about 7 = 0 to obtain

dyAt= CO = YI,o+ ar T(0) + . . .

T=0

c-IX I = (c-1Y) + (d-:-21-) T(0) + . .dt dt drdt

where r(0) denotes the value of the function T(t) (obtained by solving equation(10-103) for T as a function of t) at the point t = 0. Upon substituting in equations(10-103) to (10-105), we find

Y(t = 0) = yo (0) + e [yi (0) + (0) (ici-t)i.2)7.01 + .

dy_ (dyol r (dyi) (dyo)t, (0) (d, o)

1+dt It=0 \dr ir=0 L \ dr /7.0 \ dr /7=0 \dr2 jr=d

Then substituting these expressions into the boundary conditions (10-98) andequating to zero the coefficients of like powers of e shows that

yo(0) =-- 1dy_o (0) =0dr

yi (0) + ti (0) CAP- (0) = 0dr

(10-108)

309


dyi dy d2(0) t;(0) 0 (0) + W

dO) (0) = 0

dT dT T

o

or, using the boundary condition (10-108) to simplify the second group ofboundary conditions,

MO) =-- 0

(10-109)dyi

(0) + t1(0) dd2

dry0

(0) = 02

The solution to equation (10-106) subject to the boundary conditions(10-108) is

yo COS COT (10-110)

This has the same form as the zeroth-order solution obtained by the regularperturbation procedure. However, upon substituting equation (10-110) intoequation (10-107), the latter equation becomes

d2y1+ co2y1 cos3 COT 2C.02t1 cos COT ale,' sin (AT

dT2

= 1 cos &or + (-3 2co2t;) cos COT Ct1t7 sin COT4

We now determine the function ti (r) so that secular terms will not appear inthe first-order solution y,. These terms arise because the solutions cos curand sin COT of the homogeneous equation appear in the nonhomogeneous terms.Thus, the secular terms will not occur if we choose ti so that the coefficients ofthese two terms vanish. Hence, we take t'i'= 0, t;=3/8c.o2, or

3t1 =

8w2(10-111)

where we have set the constant of integration equal to zero in order to simplify

310


the second boundary condition (10-109). And the differential equation for yibecomes

1cd2y1 + (02y1=

4cos 3(o7

T2

Now since substituting equation (10-111) into the expansion (10-103)shows that

3ET -4- . .12

we find upon neglecting higher order terms in E that

1 t+ . . . = (1- 1,)2) t+ .

(1 +)

And when this is substituted into the zeroth-order solution (10-110), we obtain

3Eyo= cos (co 8co) t

Thus, we find that the frequency is modified by the nonlinear restoringforce even in the lowest order term of this expansion. In contrast to this, thedirect expansion leads to no information about the frequency change. Theexpansion can be carried to higher orders by continuing to use the functionstn for n = 2, 3, . . . to eliminate the secular terms. And this expansion will beuniformly valid for all times since no secular terms will appear.

10.5 MULTIPLE-TIME METHODS

Equations which contain two disparate time or length scales are frequentlyencountered in practice. This happens, for example, in problems which involvea small force acting over a long period of time. An example of this is a spring-mass system subjected to a weak viscous damping. The two times which occurin this problem are the basic period of oscillation and the damping time whichoccurs over many periods. The basic idea of the method is due to Kuzmark.

311


And a detailed discussion of this method is given by Kevorkian in reference 38.The presentation and examples used herein are taken from this report. Themethod takes into account the long-time effects in such a way as to renderthe expansion uniformly valid. As in the preceding example, it involves reason-ing about the first-order terms to determine the zeroth-order solution.

We set up the expansion in such a way that a fast time variable t and aslow time variable i are explicitly exhibited. The slow variable is assumed tobe related to the fast variable by

i=4)(e)t (10-112)

where lim =0. And we suppose that the solution to the differential.oequation has an expansion of the form

y(t; E) vo(E)fo(t*, E)+1.1()fi (t*, i) + v2(E)f2(e, I) + . . . (10-113)

where 11,;()1 is an asymptotic sequence as E -) 0 and t* is another fast timevariable related to the fast variable t in such a way that it accounts for thechange in frequency that occurs in the problem. This relation is taken in theform

t [1 +pt,i(E)oh+p,2(6)(02+ . .] (10-114)

where {pti(E)} is another asymptotic sequence and Itojl is a sequence ofconstants which are determined by the problem in such a way as to renderthe expansion (10-113) uniformly valid.

The variables e and Fin equation (10-113) will be treated as independent.Hence, the ordinary differential equation satisfied by y will be transformedinto a sequence of partial differential equations. However, it turns out thatthey can still be treated as ordinary differential equations.

Upon differentiating equation (10-113) with respect to t, we find that

( ay\ de (22.) didt kaek dt dt

But since equations (10-112) and (10-114) show that

312

dt*dt 14(e)(°'+ µz(e)c')2+

iddt= (1)(E)

this becomes

d afot*dty

[1+ ili(E)oh+ . . .]a

[vo(E) v () a*

or


4)(E) res(E)

afoai vi(E) --77a +

-1

dy fafovo(E) {cat* [1+ p,i(e)(01]+ 0(0n+ vi() a3+

This equation shows that changes which occur on the slow time scale F aresmall compared with the changes which occur on the fast time scale.

The method is best illustrated by considering a particular example. Thus,consider the equation

d2y tdy\3

t2 Y.(10-115)

which governs the behavior of an oscillator with weak cubic damping. Weshall seek an asymptotic solution which is uniformly valid for 0 t = cc, subjectto the boundary conditions

y (0) = 0 (10-116)

dy(10-117)dt

In this case, it is reasonable to begin by choosing the functions 4 and the twoasymptotic sequences {Rh and {pki} to be 4,(e) =e and vj= µj =ej. Then

313


t--= Et and the expansion (10-113) becomes

with

y (t; E)=f0 (t*,I)+Efi (t*, i ) + . .

t* = t (1 + co2E2+ . .

where we have omitted the term Emit since we wish to ensure that Et onlyoccurs in the solution 109 as t. Hence,

dy afo cfo afi , , 9\E )

dt at* at at

2y 2f0 avo 2f,Ca

2E _± Ea-4-0(E2)

dt2 at*2 at*at at*2

Upon substituting these results into equation (10-115) and equating to zerothe coefficients of like powers of E, we find that

a2f0-Ffn = 0at*2

a2fi 4_ 2 a2f*a0

C* )3+fi = 0at*2 att at

And since y(0) =fo(0,0) + Efi (0,0) + . . . and

109 There is a certain amount of arbitrariness in the choice of the variables t* and z.

314

(10-118)

(10-119)

afodt (0) a t*

afo0,0 at 0,0


afi +.at 0,0

the boundary conditions (10-116) and (10-117)

fo (0,0) =0

0) = 0 afi

afo

show

o,o

+a.foat

that

=1

=o,o

cos t*

0

(10-120)

(10-121)

(10-122)

at*

0,0

is

*+Do(i)

fl (0,at*

Now the solution to equation (10-118)

fo (t*, = Co(7) sin t

where the functions C0(t) and D0(t) are arbitrary functions arising from theintegration. But substituting equation (10-122) into the boundary conditions(10-120) shows that

D0(0) = 0 Co(0) =1 (10-123)

As in the preceding method, we now determine the functions Co and Doso that the first-order solution fi will not contain any secular terms. To this end,notice that equation (10-122) implies that

(11at3-4, 3= (Co cos t*Do sin t*)3= Co cos3 t* 3DoCg sin t* cost t*

3D8C0 sine t* cos t* sin' t*

Hence, we find, upon using the identities

315

486-942 0 - 73 - 21


cos33

3 * __

4t cos 3t-+-

4cos t*

1sin3 t*=--4 sin 3t*+ 3 sin t*4

sin t* cost t* = sin t* sin3 t*

sine t* cos t*= cos t* cos3 t*

that

(11)3 = C° °

(C°2-F D2) cos t*°

D (C2 + D2) sin t*at 4 4 ° °

1 1+ 4 Do (Do 3C8) sin 3t*+ 4 Co(C8 3Dg ) cos 3t* (10-124)

We therefore find, after subsiiiuiing equaiions (10-122) and (10-124)into equation (10-119), that

2at.6 +fi = Do (Ds + CF,) + 2 a°i sin t*4 dtCo (Ds+ cs) +

dcocos t*

*2

1

°Do(D2

°3C2 ) sin 3t*--1

°Co(C2 3D° 2) cos 3t* (10-125)4

In order to ensure that secular terms do not occur in fi, we must eliminatesin t* and cos t* from the nonhomogeneous term of this equation. This can beaccomplished by putting

dDo,-+ 32 Dowg+ =0

dt

dCo+-4

32 Co(Dg+Cf,)= 0

dt

316

(10-126)


Upon multiplying the first of these by Do and the second by Co and adding theresults, we get

which has the solution

(D a+ 2= 0= 0dt 4

D, + = 1

34

t+Ki

where K1 is a constant. And substituting this into the boundary condition(10-123) shows that we must take K1 =1 to obtain

Dg+c8,_1

t.+1

But substituting this into the first equation (10-126) shows that

2dDo + 3 Do =0di 4 3 -t + 1

(10-127)

Since the solution to this equation subject to the first boundary condition(10-123) is Do( i) = 0, equation (10-127) becomes

coV43 1+ 1

1

where the positive square root is taken to ensure that Co satisfy the secondboundary condition (10-123). Substituting these results into the zeroth-ordersolution (10-122) now shows that

317


fo(t*,

4+ 1

sin t*

Hence, we obtain the one-term uniformly valid asymptotic expansion ofthe solution to equation (10-115)

*sin t sin tY

V-3 7+1 V1+4et4

The procedure can be continued to obtain higher order terms, the solution toany given order being determined by reasoning about the next higher orderterms.

318

CHAPTER 11

Numerical Methods

It frequently happens that the differential equations encountered in prac-tice cannot be solved by the exact and approximate methods discussed in thepreceding chapters. In such cases it is often necessary to resort to numericalmethods in conjunction with a digital computer. These methods usually involvereplacing the differential equations by a number of algebraic equations, calleddifference equations, in such a way that the solution of the difference equa-tions is in some sense close to that of the differential equation. There are alarge number of numerical procedures available. The choice of method isinfluenced by the type of auxiliary conditions as well as by the form of the equa-tion. Thus, when all the auxiliary conditions are imposed at a single point(initial conditions), the solution can be developed by means of "marchingtechniques," which solve the difference equations in succession. Thesemarching solutions can be carried out either by using implicit methods such asEuler's method and the Runge-Kutta method or by using explicit methodssuch as the Adams method and the improved Euler method. Each of thesemethods has advantages and disadvantages which will be discussed sub-sequently. When auxiliary conditions are imposed at two points (boundaryconditions), it is usually possible to solve linear equations directly by usingeither the superposition principle or finite ditterence schemes which involvematrix methods. However, wnen the equations art nonlinear, it is often neces-sary either to linearize the problem or to reduce it to a set of initial-valueproblems and then use iterative or matrix techniques to obtain the solution.

The subject of numerical solutions to differential equations is quite vastand we cannot hope to cover it completely in a single chapter. For moredetailed information, the reader is referred to references 39 to 42.

319


11.1 APPROXIMATION BY DIFFERENCE EQUATIONS: ERRORS AND INSTABILITY

Consider the general system of nth-order differential equations

F04"), y"-", x) = 0

defined on the interval a x b and subject to appropriate initial or boundaryconditions.

In order to obtain a numerical solution to this system we first partitionthe interval a x 25. b. .A partition of the interval a x tc. b is defined to beany finite set of points x1, x2, . . which has the property that a = xi <X2 < < Xm+1= b. The length hi of the jth subinterval,"° xi x xi+i, iscalled the step size. Thus,

= xi+, xi foi: j= 1, 2, . . m

The system of differential equations (11-1) is then "replaced" by a setof algebraic equations, say

Gk(Yi, , ym+1; x1, . , Xtn+1) = 0 for k = 1, 2, . p (11-2)

caRed difference equations. Their solution is the set ofm + 1 vectors yt, . .

ym+i, which are approximately equal to the values taken on by the solutiony= f(x) of the system (11-1) at the m +1 points xi, . . 1...+1. Before discussingthe various methods whereby such difference equations can be constructed,we shall first consider certain types of errors which can occur when differenceequations are used to obtain numerical solutions.

The discretization or truncation error E1 at the ith step is defined to bethe magnitude of the difference between the solution to the differential equa-tions at the point xi and the solution yi of the difference equations. Thus

E.= f(xi) y1{

This error depends only on the type of difference equation used and is inde-pendent of the method by which it is solved.

u° It is not necessary to have all hj equal, but it is usually desirable.

320

NUMERICAL METHODS

However, there is also an error which is caused by the numerical procedureitself. Because a computer can accommodate only a limited number of sig-nificant figures, it cannot store accurately an irrational number or even arational number requiring precision beyond the computer's capability. There-fore, the computers themselves introduce an error which results from thenecessity of rounding off numbers: it is known as the round-off error. Theround-off error at any step in the computations propagates to the next step andis combined there with the round-off error of that step. The generation of theround-off error at each step is extremely unpredictable. Precisely for thisreason, analyses of round-off error often treat the error per step as a randomvariable (see ref. 43).

Closely related to the question of error is the question of stability, whichmust be considered in certain instances before a numerical solution canobtained. Various types of instability can arise. If the instability is inherentin the differential equation itself, it is called an inherent instability. For ex-ample, consider the initial-value problem

(11-3)

subject to the initial conditions yi (0) =1 and y2(0) = 10. The general solutionfor equations (11-3)

y, (x) = C2e1OX

Y2(x) = 10Cie-oz + 10C2eioa-

And the initial conditions ate satisfied by taking C, = 1 and C2 = 0. However,when a numerical procedure is used, it will usually be impossible to satisfythese initial conditions exactly. But a small error in determining the constantC2 will then allow the second terms in the solutions to eventually grow so largethat they will dominate the first terms, which correspond to the solution beingsought. Situations of this type arise most frequently when the initial-valueproblem is being solved as part of an iteration procedure to solve a boundary-value problem.

321


Instabilities can also arise from the difference equations (even when thedifferential equations are stable). They are then called induced instabilities.These instabilities can result in spurious solutions to the difference equationswhich do not correspond to solutions of the differential equations. For moredetails the reader is referred to reference 43.

11.2 INITIAL -VALUE PROBLEMS

In this section we shall consider certain types of "difference schemes"which are suitable for obtaining numerical solutions to initial-value problems.

11.2.1 Explicit Methods

11.2.1.1 One-step processes: Taylor series method. It has been indicatedin chapter 3 that any normal system of ordinary differential equations canalways be written as a first-order normal system, which in vector notation hasthe form

dydx

t.(x, y) (11-4)

We shall now consider some methods, referred to as one-step processes,for obtaining numerical solutions to this equation on an interval a x bsubject to the initial conditions y= yi at x= a. For any given partition ofa b, say x1, . . zin+1, we can seek to approximate the solution y= f(x)of the system (11-4), subject to these initial conditions, by replacing thisequation by the set of algebraic equations (or difference equations)

yi4.1= yi+ G(zi, yi)hi for j= 1, 2, . . m (11-5)

It can be seen that, starting with the value yi, equation (11-5) can be used tocalculate yj successively at the points j =2, 3, . . m + 1. We hope thatthe vectors yj will provide good approximations to the values f(xi) of the solu-tion to equation (11-4) at the points

The discretization error which occurs when using a difference equationto integrate a differential equation across a single step, T(xj, 10, is called thelocal truncation error or the truncation error per step. Thus, the truncationerror per step is the error induced by using equation (11-5) to calculatef(xj+ hi) approximately from the value f(xj) = yj or, symbolically,

322

NUMERICAL M2THODS

T(x.i., hi) yi+2 f(xi + hi)

Hence, upon substituting in equation (11-5) with yj= f(xj), we get

T(xi, hj)=-. f(xi) f(xj+hi) + G (xi, f(xj))hi

But Taylor's theorem shows that "I

f(xj+ hi) = f(xj) + ),,j h.i+ 0(hi2)

Hence,

T(xi, hi) = G(xj, f(xi )) ( I hi + 0(hpdx fxj I

But since, by hypothesis, f(x) satisfies the differential equation (11-4), thefirst term vanishes arid we obtain

T(xj, hi) = 0(k?) as hi-0 0 (11-6)

More generally, we can attempt to approximate the solution to equation(11-4) by means of a difference equation of the form

yi+i= yrj+ 43(xi , yi; hi)hi for j= 1, 2, . . m (11-7)

where the function 1 is to be chosen so that in some sense the solutions toequation (11-7) provide good approximations to the solutions of equation(11-4) for sufficiently small step size hi. In order that this be the case, wecertainly must require that

y; h ) --) G(x, y) as h 0 (11-8)

If we again let y= f(x) be a solution to equation (11-4), the truncationerror per step T(xi, hi) is

"' The concept of order and the symbol 0 are introduced in chapter 9.

323


T(xi , hi) = I f(xi ) f(xi + hi ) + (I)(xi , f(xi ) ; hi )hi I

The same argument as was used in obtaining equation (11-6) %Ned to-gether with condition (11-8) now implies that

T(xj; hi) = o(hi) as hi 0 (11-9)

This shows that the truncation error per step goes to zero faster than themesh size hi. Other things being equal, it is of course desirable to have thetruncation error per step approach zero at the fastest possible rate as hi-0 0.In order to have some measure of this rate we define the order of a givendifference scheme to be the largest number p such that

T(xj; hi) =-- 0(hr 1 ) as hi-0 0

Equation (11-6) shows that the order of the difference scheme (11-5) is 1.And, more generally, equation (11-9) shows that the order of any differencescheme of the type (11-7) which satisfies condition (11-8) is greater thanzero. The numerical method corresponding to the difference scheme (11-5)is called Eater's method. Although this method is very simple, it is prone toround-off errors and is therefore infrequently used.

The truncation error per step can be used to obtain a bound on the (cumu-lative) truncation error for difference equations such as equation (11-7), thatis, difference equations which determine the solution at the point xi+, onlyin terms of quantities from the preceding step. Instead of considering thegeneral system (11-4), we consider, for simplicity, only the single differentialequation

dy= G(x, y)

dx

Suppose G(x, y) satisfies the Lipschitz condition with respect to y

IG(x, G(x, ) 1 AllY

324

and let

A maxT(xi, hi)

hi

NUMERICAL METHODS

for j= 1, 2, . . m

Then it is shown in reference 4 (p. 181) that the truncation error Ei is, at most,

that is,

A (elximrilm 1)

E. A (elx.rxilm 1)M

Now for any difference scheme of order p, there exist constants Di inde-pendent of the mesh size hi such that

T(x.D.hP

h;

And if we put

E= max hjfor j= 1, 2 . . m

D max pi

then L D and the truncation error satisfies the inequality

L.1 MDEP

for j= 1, 2, . . m

which shows that the (cumulative) truncation error is of order 1' when thedifference scheme is of order p.

325


In principle, it is easy to derive formulas for numerical integration ofthe system of equations (11-4) which are of an arbitrarily high order. Thiscan be accomplished, for example, by the method of Taylor's series. Instead ofapplying this method to the general system of first-order equations (11-4),we shall again, for simplicity, consider only the single first-order equation(11-10). There is no difficulty in extending the ideas to the general system(11-4).

Let y=---- f (x) be a solution to equation (11-10) and let g(x, y) be any mimescontinuously differentiable function of x and y. Taking the total derivative ofg with respect to x along the curve which is obtained by plotting y=f(x) gives

dg ag ±ag dy ag Gagdx- ax ay dx ax ay

Applying this formula to the function g1(x, y) gives

dX2 dx axd2g dgi gl agi (ax

aya a ) (ax a al alt

g2±

ay G + G ay gax

+ Gayl

g

and, in general, we obtain

ndng dgn-i a gi G + G gay ax aygn dxnfor n = 1,2, . . r

dx a x

Thus, in the special case when g(x, y) y), this equation becomes, inview of the differential equation (11-10),

d2vGs+ GGy

dx2

d3Y =-- 2GG,y+ G2Gyy+ GsGy+ GGr,dx3

326

NUMERICAL METHODS

dn+lydxn+'

a a

ax+ G

ayG

dr+1Y G Gdxr+' ax ax

Now suppose that G is p- 1 times continuously differentiable and put

(x, y; h) k." hn a+ G a )71G 11-14)((n+1)! ax ay

Then, at least in principle, (F can be calculated for any integer p simply bydifferentiating the given function G. Hence, when y=f(x) is a solution to equa-tion (11-10), it follows from equation (11-13) that

(1)(x, f(x); h) hn cla÷ly(n+1)! dx"'

Thus, the truncation error per step, which is incurred when the differenceequation

yj.,_!----yj±(1)(x3, 35; Miti for j=- 1, 2, . . m (11-16)

is solved to obtain an approximation to the exact solution, y=f(x), of thedifferential equation (11-10), is

T(xi, hi) = I fixi)-fixi+ hi) + f(xi); hi)hi

But inserting equation (11-15) shows that

13-1 hp +iT (xi, hi) = I f(xi) f(xi+ hi) + E

(n+ 1) ! dxn+1 .n=0

n97

(1,


Hence, upon applying Taylor's theorem we find that

T(xi, hi) 0 (hrl)

which shows that the formula (11-16) with 1:1) determined by equation (11-14)is of order p. The method, of Taylor's series is efficient for linear systems oreven for equations where G is a polynomial of low degree in x and y. However,as can be seen from equations (11-11) and (11-12), the method usually becomesextremely complex. In order to avoid this complication due to the successivedifferentiation and at the same time to preserve the increased accuracy whichis afforded by using the Taylor's series method, a technique introduced byRunge, Kutta, and Heun known as the Runge-Kutta method can be employed.

In this case, the function 4:13 in equation (11-7) is taken to be of the form

where

and

43(Xi, Yi; hi) = E asks8=1

G(xj, yi) (11-18)

8-11E8 = G (xj+ gshi, yi+ hi E Xs - 1, n kn) for s =2, 3, . . r (11-19)

n=--1

For any given integer r, the parameters a8, As, and As, are to be determined insuch a way that the order p of equation (11-7) is as large as possible.

For simplicity, we shall again restrict our attention to the single first-orderequation

dydx.G(x, y) a b

In this case, equations (11-7) and (11-17) to (11-19) reduce to

328

( 11-20 )

Yi+i= 43(xi, yj; hi)hi for j= 1, 2, . . m (11-21)

ks= G (xi+ gshi, yi +hi

44)=--E ask.,=,

ki=G(xj, xi)

8_,Xs-1, n kn)

n=1

NUMERICAL METHODS

(11-22)

(11-23)

for s =2, 3, . r (11-24)

And if y=f(x) is a solution to equation (11-20), the expression for T(xj, hi),the truncation error per step, reduces to

T(xj, 10= if (xi) f (x; + + f (xi); /Oki (11-25)

Applying Taylor's theorem to the function T(xi, hi) gives

P 1 [amT(xi, h)T(xi, hi) =m

I 1n + o(hri)ahm h=0

hm=0

But differentiating equation (11-25) m times gives

ramr(xj, h)1L ahm jh=0

dray171 hT-1 h=0 WX7n) s=sIJ

Hence, the method will be of order p, provided that

tam-10 (c/my) f 0 for m= 1, 2, . .

m alen-1 h=O \delix.r Lp ci for m= p+ 1J J

P (11-26)

If we substitute equation (11-13) and equations (11-22) to (11-24) into equation(11-26) and equate to zero the coefficients of all the independent partialderivatives of G, we will get a set of nonlinear equations for as, p,,, and As,n.For any given value of r, there will be a largest value of p for which these equa-tions can be solved. For 1 r 4, this value of p turns out to be equal to r.

There is a certain arbitrariness in the solutions of these equations for theconstants as, gs, and As,n. Thus, for r =2, one of these constants can be chosenarbitrarily; and for r= 3 and r= 4, two of the constants can be chosen arbitrar-ily. The computations are carried out for the general case in reference 41.

329


Here we shall merely illustrate the method by considering the case wherer= 3. Then equations (11-22) to (11-24) reduce to

alkj a2k2 + a3k3 (11-27)

kl= G(xj, xi)

k2= G(x.i+ g2hi, xi+ h;XI,' ki) (11-28)

k3= G(x.i+ 12,31ii, xi+ h;X2, ki + 2 k2)

Expanding k1, k2, and k3 in a Taylor series about h; =0 and retaining termsonly up to hi give

k1= G (xj,

k2= G xi) + hi (PaGs+ Xi, iGGy) x=sju= b;

+h.7 Gsx+ PaXi,iGGsy+ +_m,IG2Gyy) 4_0(h3)ft: ifj

01-29)

k3 = G (xi, Yi) 11.; (g3Gx+ A2, iGGy+ A2, 2GGy) x.=.vY=1/j

+hJ 12AG. tk3 (A2, 1+ A2, 2) GGsy + 2 (A2,1+ A2, 2)2G2(;a/

+ A2, 2 (142Gx+ Al, 1GGy) 4=a:1+ 0(N)y=1/j

330

Now it is clear that

1 /3-41\

J-m! kahTlh.0

NUMERICAL METHODS

is the coefficient of (hi) "' in the Taylor series expansion of 10 about hi= 0.Hence, when equations (11-29) are substituted into equation (11-27), thefunctions

1 (a- "I1m!kahrhi.0

for m= 0, 1, 2 are simply the coefficients of itY , hi, and 10c

respectively, in theresulting expression. Thus,

(I) (xi, yi; 0) (ai + a2 a3) G yi) (11-30)

X1. 'GC Y) s= x j+ a3 [1.1,3Gx + (X2, 1 ± X2, 2) erGis].r=xj (11-31)(M3

a2 (g2Gx+11=11j 11=11j

1 024)\a2 PdXi, IGGry + Xf, igGyy)kahphi=0

a3 [ A3G.r.r+ /13 (X2, 2) GG.ry + 2 (X2. 1+ X2, 2 )2G2GY

+A2,2 (i.L2Gx+ IGGy) (11-32)x=xi

On the other hand, equations (11-26) show that

331

488-942 0 - 73 - 22


(I) (xi, yi; 0) = (2)z=zi

la(1)1 (d2Y\kaki) hro Vir

3 (q),,,..=(3,4--.x.Y),..,

Upon substituting equations (11-11), (I1-12), (11-20), and (11-30) to (11-32)into these relations, we find that the resulting equations are satisfied identicallyin xi and yi only if the constants as, gs, and As,. are chosen so that the coeffi-cients of all the independent derivatives of G vanish. This will occur if, and onlyif, the constants satisfy the following algebraic equations:

a1+ a2+ a3= g2a2+ pL3a3=--2

1Xi ,ia2 + A2,1+ pia2+14a3= 1

2 3

1g2X1,1a2 ± P.3 ( X2,1 4- X2,2 ) a3

1xija2+ (A2,3+ X2,2)2a3= 3

1 1/12X2 ,2a3 6 X1,1 X2 ,2a3 = 6

These equations imply that

XI ,1 = (12 X2,1 ± X2,2 = IL3

and there are four independent equations which must be satisfied by the re-maining six unknown constants. Hence, two of these constants can be chosenarbitrarily.

The most frequently used Runge-Kutta method is of the fourth order.The values of the constants a,, As, and for the general vector equations

332

NUMERICAL METHODS

Table 11-1. Choice of Parameters forFourth-Order Runge-Kutta Method

ParametersStandard

RungeKuttamethod

kutta's method

al 1/6 1/8as 1/3 3/8as 1/3 3/804 1/6 1/8

Ai 1/2 1/3Aia 1/2 2/3As 1 1

AI., 1/2 1/3A2,1 0 1/3Asa 1/2 1

As,, 0 1

Ass 0 1AI li 1 1

(11-17) to (11-19) for the fourth-order Runge-Kutta method are listed in table11-1 for two choices of the arbitrary constants, and because of its importancethe complete formulas for the standard Runge-Kutta method are also listed:

hin+i-=-yi+-'-6 (ki+2k2+2k3+k4)

G(xj, xi)

1 1k2=-- G xi+2

h -1 kik,2

-I 21 1 hky +-2 2

1(4= G(xj+ hi, xi+ hjk3)

There appears to be only a slight advantage which can b,; gained by changingthe choice of the arbitrary parameters.

333


Although the Runge-Kutta melhod involves fairly simple formulas, ithas certain disadvantages. Thus, (1) the method is limited to the fourth orfifth order; (2) if the function G is complicated, the evaluation of the k, fors= 1, 2, 3, 4 at each mesh point can be vite time consuming; (3) it will calcu-late a solution even at points of discontinuity without giving any indication thatthis has been done; and (4) there is no readily obtainable error analysis.

The lack of any error analysis for the fourth-order Runge-Kutta methodcan be partially compensated for by using certain rules of thumb. Thus, forexample (see ref. 41), if the quantity

k2 k3klk2

becomes much larger than a few hundredths at any point xj, the step sizehj should be uecreuseu.

11.2.1.2 Multistep processes: Finite differences. Up to this point we havebeen discussing one-step difference equations, that is equations which deter-mine the value of the dependent variable at the step xj+, completely in terms ofits value at the preceding step X. There are other types of difference equationswhich can be used, called n-step equations,"2 which utilize the values of thedependent variable at the first n preceding steps, say xj. xj_i, . .

to determine its value at the step xj+1.Before discussing these difference equations it is first convenient to in-

troduce the concept of difference operator. Thus, the difference operatorsA, V, and 8 corresponding, respectively, to the forward difference, the backwarddifference, and the central difference are defined by

Ay(x) y (x+ h) y(x)

Vy(x) a y(x) y (x h)

Sy' (x) = y + y

If the function y(x) is defined only on a finite set of points, say x, < x2 < . .

< xm+i with xj+, h for j-=-- 1, 2, . . m, we shall sometimes write

112 The associated numerical procedure is referred to as an n-step process.

334

NUMERICAL METHODS

yj y(xj)= y (xi+ (j -1) h)

Then the notation for the first two difference operators becomes

A.r.i= Yi+i- V Yi -1

These operators arise in approximating the derivative of functions. Thenature of this approximation can be seen from the relations

limAv(x) V v(x)

rnSy(x) dy(x)

,h

= lih-00 n 11-.4) n h--.0 2h dx

Applying these operators twice in succession gives the second differences

A2Y(x)=A [Ay(x)]=P [y(x+h )-y(x)] (.1t +2h) -2y (x+ h)+ y(x)

V2).(x)= y(x) -2y (x- h) + h (x-2h)

82y(x)=y (x+h) 2y (x)+y (x-h)

These formulas can be used to provide approximations to the second deriva-tive since

d2y(x)lim

Azy(x)rn=liv21.(x) =lint

821,(x)

dx 4-0 h2 h2 h_.0 h2

There are various manipulations that can be performed with these and otherdifference operators which are sometimes useful for obtaining finite differenceequations from differential equations. A fairly detailed discussion of this isOven in Hildebrand (ref. 44).

Now consider the general nth-order differential equation in normal form

dny dy dn.-1y\(IX" 1\X " dx1L-1)

a x b (11-33)

335


Suppose that the interval a x b has a partition x1, xs, . . for whichall the subintervals xi x xj+i have the same length h. Upon replacingeach derivative (dky)I(dxk) for k= 1, 2, . . n in equation (11-33) by itsforward difference approximation h-k Akyi, where yj is the approximation toy [a + (j-1) Id=y (xi), we obtain the difference equation

1 AAn)) = h nG (Xj, yj, -h- Lot),

1

hn1 (11-34)

However, since Alkyl is a linear combination of yj, xi+i, , Yi+k, this equationis essentially of the form

Yi+m=4)(i, . , Xi+m-i; h)

which is clearly an n-step difference equation.In fact, even first-order normal differential equations can lead to n-step

difference equations. Thus, consider the differential equation

dydx=G(x' y)

and approximate the derivative by the central difference (yi+i yi_1)12h toobtain the difference equation

yj+i--=x1-1-1-2hG(xj, xi)

which is clearly a two-step equation.Notice that, in order to start the solution of an n-step method, we must

have a prior knowledge of the values of y1, Y2, Y39 ., YrI9 that is, the values(or approximate values) of the solution at the first n mesh points. A one-stepmethod requires only a knowledge of the initial value y1. Hence, when n > 1it is in many instances necessary to introduce some auxiliary method for deter-mining these values. The one-step methods are, therefore, said to be self-starting. In a one-step method, the mesh size h1= xj+z xi can be varied ateach step. With a multistep method, it must usually remain fixed.

336

NUMERICAL METHODS

11.2.2 Implicit Methods: Predictor-Corrector

In the methods discussed so far., the value of the dependent variabley2 +1 at the step xi+, is determined explicitly in terms of its values at one ormore preceding steps. We can, therefore, calculate the values of the dependentvariable recursively. Hence, such methods are called explicit methods. Thereare other methods, however, in which the formulas for calculating yi+i fromthe values of the dependent variable at the preceding steps are not solvedexplicitly for yi+i but determine this variable only implicitly. Such methodsare, therefore, called implicit methods.

In order to see how difference equations of this type arise, let

a = x1 <x2 <. . . < xm+1=

be a partition of the interval a b; suppose that each subinterval xi '45 xb

xi+, has the same length, say h; and consider the integral jf f(x)dx. Recalla

that this integral can be evaluated numerically in an approximate fashion byusing either the trapezoidal rule

ni 1f(x)dx --,,[f(xj,i)+f(xinhj= 4.

or by using Simpson's rule

Jo i=

n 1 1Ef (xj+2) + 4f (xj+1 ) + f Ckinh

Now, consider the first-order differential equation

dydx

y) (11-35)

Integrating both sides of this equation first between xi and xi+i and then betweenxi and xi+2 gives, respectively,

337


and

xj+1yi+1=yri- G(x, y(x) )dx

x.

2

Yj + 2 -'=" Yj G(x, y(x))dx

(11-36)

(11-37)

Upon using the trapezoidal rule for evaluating the integral in equation (11-36)and Simpson's rule for evaluating the integral in equation (11-37), we get thefollowing finite difference equations for approximating the solution to equation(11-35):

and

Yi+ [G(xj+1, G(x.i, yin2

(11-38)

Y.i+ [G(xj+2, )'i+2)+ 4G(xj+i, Yi+1)+ G(xj, (11-39)

Notice that in the first of these equations, ypi-i appears not only explicitlybut is involved implicitly through C on the right side. In general, it will not bepossible to solve this equation to obtain an explicit formula for Similarremarks, of course, apply to the second equation. If, instead of treating thesingle first-order normal equation (11-35), we consider the general normalsystem

dy =dx

G(x, y)

the same arguments would lead to the difference equations

yp+i = yi+ [G(xj+i, yi+1) + G(xj, xi)]

338

(11-40)

(11-4.1)

and

NUMERICAL METHODS

Yj+2 = yi + [G(xj+2, xj+2)+4G(xj+i,h

(11-42)

Notice that, in equation (11-41), xi+, is determined completely in terms ofits value at the preceding step; whereas, in equation (11-42), )7)+2 is deter-mined in terms of the values of the dependent variable at the preceding 'kwosteps. Since these formulas cannot usually be solved explicitly for the de-pendent variable, it is usually necessary to resort to an iteration process ateach step to find this variable. Thus, let

U (n+1) yi+ yi:-1)+G (xJ,

Then equation (11-41) can be written as

= (11-43)

Suppose that by some means a fairly good guess at the solution n+, of equation(11-41) can be made, say a. Substituting this into equation (11-43) givesa better approximation x;',.), to the solution, given by yWi = U (yA).

Proceeding in this manner, we obtain the sequence of approximations

v(i)i=u (yy_)4), u (y) v(3) =U ( ) , . . .+

which we hope will converge fairly rapidly to the solution of equation (11-41).A good choice of the initial approximation yp, can be obtained by using Euler'sformula, equation (11-5), to obtain

y?+), = xi+ G (xi, xi) h (11-44)

In practice, instead of solving equation (11-41) accurately for xi+, byperforming many iterations, we can obtain the same accuracy with much lesswork by taking a finer mesh size It and performing only one or two iterations.If only one iteration is performed, the method is called the improved Eulermethod. In this case, we first compute the vector Yi+1, called the predictor,from the formula

339


Y; -1-1= yj+ G (xj, yj) h (11-45)

called the predictor formula, and then substitute it into the formula

[G (Xj4-19 10+1) + G n)1 2 (11-46)

called the corrector formula, to determine yj+/. Thus, in effect, yj+1 is cal-culated from yj in two steps instead of one. Of course, we can apply the sameprocedure 'to equation (11-42). This leads to Milne's method.

These two methods are examples of the predictor-corrector methods.Other predictor-corrector methods differ from these only with respect to thepolynomial interpolation formulas from which the predictor and correctorformulas are derived. A commonly used predictor formula is the Adams-Bashforth formula (see ref. 45)

Yi+i= yi+ 24 (55G3 59G3-1 + 37Gj_2 9Gj-3)

where we have put G.; = G(xj, yj). This formula is most frequently used inconjunction with the Adams-Moulton corrector formula (see ref. 45)

+ 7271 [9G(x.i+i, Yp-1) + 19G; 5G;_i + Gi_2]

These formulas have a higher order of accuracy than the Euler's formulas. How-ever, they are not self-starting. Also, unlike the Runge-Kutta method theycannot be easily used alone with a variable mesh size. These difficulties arefrequently alleviated in practice by using the Runge-Kutta method to obtainthe starting values and also to compute the solution for the first few meshpoints after the step size has been changed. However, the predictor-correctormethods can, in the case of complicated equations, result in a considerablesavings in computer time over the Runge-Kutta method. In addition, it isusually possible with predictor-corrector methods to monitor the error as thecalculation proceeds.

Another difficulty with the predictor-corrector methods is that in somecases, they are subject to certain types of instabilities which do not occurwhen the Runge-Kutta method is used. This instability manifests itself first by

340

NUMERICAL METHODS

resulting in an error which is larger than expected; and when one attempts toreduce this error by decreasing the step size, the error actually increases. Amore detailed but elementary discussion of this instability is given in reference45.

11.3 BOUNDARY-VALUE PROBLEMS

11.3.1 Linear Equations

11.3.1.1 Use of superposition.The methods discussed previously have,all been methods for solving initial-value problems. However, in the case oflinear equations, these methods can be used in conjunction with the super-position principle to obtain solutions to boundary-value problems. Thus, inorder to solve a boundary-value problem for a linear differential equation, weneed only solve numerically the same number of initial-value problems asthe order of the differential equation, provided these problems are chosen insuch a way that their solutions are linearly independent of one another. It iseasily seen from section 1.6 that this can always be done by choosing theinitial conditions of these problems so that they have a nonzero Wronskiandeterminant. Then any boundary-value problem can be solved by forming alinear combination of these solutions with the constants adjusted numericallyto satisfy the imposed boundary conditions. This is a particular example of howsome a priori knowledge of the properties of the solutions of the equations tobe solved can be utilized to simplify the numerical procedure for obtainingthese solutions.

11.3.1.2 Finite differences.Another method for solving linear boundary-value problems is the method of finite differences. In order to illustrate thismethod, let us consider the second-order linear equation

dy p(x) y + q(x)y= r(x) a b (11 -47)

subject to the boundary conditions

y(a) = A

y(b) = B(11-48)

341


Let a = xo < xi < . . . < b be a partition of a x b with equalmesh size h.

Upon approximating the derivatives by appropriate central differences,we obtain the following difference-equation approximation to equation (11-47)

yi+i 2n+2h 2h

pi+ y i q i = rj for j= 1, 2, . . . m

where we have put pi = p(xj) , q; = q(xj) , and r; = r(xj). This can be writtenas

h(1- -h p)y.._i+ (h2qi 2) yi + (1 + p) y-4.1 = h2ri2 3 ' ' ' for j= 1, 2, . . ., m

Upon using the boundary conditions (11-48) to replace yo by A and yoz+i byB, we obtain the following set of m equations in the m unknowns yl. y2.. .

Ym:

(h2qi -2) yi + (1 + Pi) Y2 h2ri + 1)A

h(1

2p2)yi+ (h2qi 2) y2 + -r- P2) Y3 = h2r2

-1 m-2(1 Pm -1 V + (h2qm-1 2) ym_i+ (1+ -h n2 ,m-i) ym= h2rm_i

-h pm) ym_i + (h2qi- 2) Ym= h2rm + 2 pm) B2

This equation can be written in matrix form as

My= c (11-49)

where y is the vector y= (yi, y2, ., ym), M is a matrix of the form

342

NUMERICAL METHODS

PI l'I 0 0 . . . . 0 . 0 0 0a2 ,G2 y2 o . . . . o 0 0 00 ft,-3 P3 y3 . . . . 0 0 0 0

M=

. . . . 0 am -1 gm-1 'Ym-i

. . . . 0 0 am gm

0 0 0 00 0 0 0

and c is a vector. The entries in M and c are all known since p, q, and r areknown functions. For obvious reasons, the matrix M is said to be tridiagonal.Tridiagonal matrices can be numerically inverted easily and quickly; and,therefore, equation (11-49) can readily be solved by inverting M to find thesolution vector y. A particularly convenient technique is known as the lineinversion method. In setting up finite difference problems it is not alwayspossible to obtain tridiagonal or even n-diagonal matrices. However, wheneverpossible, the difference equations should be set up to obtain n-diagonal matricessince they can usually be inverted more easily than other types of matrices.In addition, when the system is programed for a digital computer, it is notnecessary to define all m2 locations of the coefficient matrix M; in the case ofa tridiagonal matrix, for example, only 3m locations need be allocated to Mwhile performing the inversion. In any case the matrices which arise in thefinite difference methods may usually be inverted by either implicit or explicitmeans. For a discussion of the various methods for accomplishing this, thereader is referred to references 46 and 47. It sometimes happens, however,that it is not possible to invert these matrices; it is then necessary to use aniterative process to solve the matrix equation (see ref. 47).

11.3.2 Nonlinear Equations

11.3.2.1 Shooting methods.Boundary-value problems for nonlinearequations can be solved by using "shooting methods." To use these methodsthe problem is first transformed to an initial-value problem by guessing enoughadditional initial conditions at one boundary to allow the integration to proceedacross the interval to the other boundary. In the initial trial the specifiedboundary conditions at the second boundary are unlikely to be met. But, byadjusting the additional initial conditions imposed at the first boundary, it is

343


possible to come closer to -satisfying the prescribed boundary conditions bycarrying through a number of iterations.

The required adjustments to the additional initial conditions can bemade in a number of different ways. Perhaps the most common of these islinear interpolation. In order to illustrate the ideas involved, consider a two-point boundary-value problem for a second-order system on the intervala x - b in which the boundary conditions y(a) = A and y(b) = B arespecified. And suppose that the initial slope y' (a) is to be adjusted until thesecond boundary condition is satisfied. When two values of the initial slopeA (a) and y2 (a) have been found which lead to the two boundary values yi (b)and y2(b), respectively, such that

Yi(b) = B y2(b)

the next trial value of y' (a) is deiermined by linear interpolation by using theprescription

y' (a) A(a) B yi(b)

Ya) Yi(a) Y2(b) MOIf this new value of y' (a) leads to a value y(b) which is either smaller or largerthan B, it can be used together with either y2(b) or yi(b), respectively, torepeat the process. The process can be continued until the second boundarycondition is satisfied to within the des5red accuracy.

One difficulty with shooting methods which sometimes occurs is that thedifferential equation is so unstable that it "blows up" before the initial-valueproblem can be completely integrated. In such cases the process of quasi-linearization can be used.

11.3.2.2 Quasi-linearization.Nonlinear boundary-value problems canalso be solved by reducing them to linear problems by a process of quasi-linearization. This process consists of replacing the original equation by asequence of linear equations in such a way that the sequence of solutions tothese equations converges to the solution of the original equation. Thus,consider the first-order normal equation

dy

dx= G(x, y)

344

(11-50)

NUMERICAL METHODS

If y is close to y, we might anticipate from Taylor's theorem that

G(x, y) G(x, y) +aT, (x, y)(yy)

We, therefore. replace equation (11-50) by the sequence

dy4" +=--- G (x, y(n) ) a

aG(x, n) ) ryn+ I) y(n)]

dx for n = 0, 1, 2, . .

of linear equations for y<n+0. To use these equations we, first, choose a reason-able guess for y")), calculate ,y41), insert it in the right side, and calculate y42).Proceeding in this manner, we obtain a sequence of solutions y4") which wehope will converge to the solution y of the original problem. Since each equa-.lion (11-51) is linear, it can be handled by the methods described previously.These ideas are easily generalized to the first-order normal system

dy =dx G(x, y)

and, therefore, to an normal system tilferential equations. The basicreference on the quasi-linearization proci:zt. as Bellman and Kalaba (ref. 48),in which are discussed various conditions which can be imposed on the func-tions G to ensure that the iterations converge.

Although all the most commonly used methods are described in thischapter, it is impossible in a single chapter to touch upon all the availabletechniques. For a more cornpfete treatment of the subject, the reader is referredto the references cited.

345/34 (

REFERENCES

1. STAKGOLD, IVAR: Boundary Value Problems of Mathematical Physics.Vol. I. Macmillan Co., 1967.

2. FRIF,DMAN, AVNER: Generalized Functions and Partial Differential Equa-tions. Prentice-Hall, Inc., 1963.

3. INCE, EDWARD L.: Ordinary Differential Equations. Dover Publications,1953.

4. BIRKHOFF, GARRETT; and ROTA, GIAN-CARLO: Ordinary Differential Equa-tions. Ginn and Co., 1962.

5. GREENSPAN, DONALD: Theory and Solution of Ordinary DifferentialEquations. Macmillan Co., 1960.

6. KAPLAN, WILFRED; Advanced Calculus. Addison-Wesley Publ. Co., Inc.,1952.

7. KAPLAN, WILFRED: Ordinary Differential Equations. Addison-WesleyPubl. Co., Inc., 1958.

8. GOLDSTEIN, MARVIN E.; and ROSENBAUM, BURT M.: Introduction toAbstract Analysis. NASA SP-203,1969.

9. MORLEY, F. V.: A Curve of Pursuit. Am. Math. Monthly, vol. 28, 1921,pp. 55-93.

10. DAVIS, HAROLD T.: Introduction to Nonlinear Differential and IntegralEquations. Dover Publications, 1960.

11. PAINLr,vt, P.: On Differential Equations of the Second and of HigherOrder, the General Integral of Which is Uniform. Actr Math., vol. 25,1902, pp. 1-85.

12. HERBST, ROBERT T.: The Equivalence of Linear and Nonlinear Differ-ential Equations. Proc. Am. Math. Soc., vol. 7,1956, pp. 95-97.

13. GERGEN, J. J.; and DRESSEL, F. G.: Second Order Linear and NonlinearDifferential Equations. Proc. Am. Math. Soc., vol. 16,1965, pp. 767-773.

14. PINNEY, EDMUND: The Nonlinear Differential Equation yil-p(x)y+cy-3= 0. Proc. Am. Math. Soc., vol. 1,1950, p. 681.

15. RAINVILLE, EARL D.: Intermediate Differential Equations. Second ed.,Macmillan Co., 1964.

347

488-942 0 - 73 - 23


16. MURPHY, GEORGE M.: Ordinary Differential Equations and Their Solu-tions. D. Van Nostrand Co., Inc., 1960.

17. KAMKE, E.: Differentialgleichungen, Liisungsmethoden and Liisungen IGewiihnfiche Differentialgleichungen. Akademische Verlagsgesell-schaft Becker & Erler kom.-ges., Leipzig, 1943.

18. CHURCHILL, RUEL V.: Complex Variables and Applications. Second ed.,McGraw-Hill Book Co., Inc., 1960.

19. AHLFORS, LARS V.: Complex Analysis. McGraw-Hill Book Co., Inc., 1953.20. CARRIER, GEORGE F.; KROOK, MAX; and PEARSON, CARL E.: Functions of

a Complex Variable. McGraw-Hill Book Co., Inc., 1966.21. NEHARI, ZEEV: Conformal Mapping. McGraw-Hill Book Co., Inc., 1962.22. MORETTI, GINO: Functions of a Complex Variable. Prentice-Hall, Inc.,

1964.23. RAINVILLE, EARL D.: Infinite Series. Macmillan Co., 1967.24. POOLE, E. G. C.: Introduction to the Theory of Linear Differential Equa-

tions. Dover Publications, 1936.25. WHITTAKER, EDMUND T.; and WATSON, GEORGE N.: A Course in Modern

Analysis. Fourth ed., Cambridge Univ. Press, 1927.26. ERDELYI, A.; MAGNUS, W.; OBERHETTINGER, F.; and TRICOMI, F.: Higher

Transcendental Functions. Vol. 2, McGraw-Hill Book Co., Inc., 1953.27. RAINVILLE, EARL D.: Special Functions. Macmillan Co., 1960.28. BUCHHOLZ, HERBERT: The Confluent Hypergeometric Function with Spe-

cial Emphasis on Its Applications. Springer-Verlag, 1969.29. WATSON, GEORGE N.: A Treatise on the Theory of Bessel Functions.

Cambridge Univ. Press, 1922.30. MCLACHLAN, NORMAN W.: Theory and Application of Mathieu Functions.

Dover Publications, 1954.31. TITCHMARSH, EDWARD C.: The Theory of Functions. Second ed., Oxford

Univ. Press, 1939.32. ERDELYI, A.: Asymptotic Expansions. Dover Publications, 1956.33. LANGER, RUDOLPH E.: On the Asym_ptotic Solutions of Ordinary Differ-

ential Equations, with an Application to the Bessel Functions of LargeOrder. Trans. Am. Math. Soc., vol. 33,Jan. 1931, pp. 23-64.

34. LANGER, RUDOLPH E.: On the Asymptotic Solutions of Differential Equa-tions, with an Application to the Besse' Functions of Large ComplexOrder. Trans. Am. Math. Soc., vol. 34, July 1932, pp. 447-480.

348

REFERENCES

35. LANCER, R. E.: The Asymptotic Solution of Ordinary Linear DifferentialEquations of the Second Order, with Special Reference to the StokesPhenomenon. Bull. Am. Math. Soc., vol. 40,1934, pp. 545-582.

36. FRIEDRICHS, K. 0.: Special Topics in Fluid Mechanics. New York Univ.Press, 1953, p. 126. Special Topics in Analysis. New York Univ. Press,1954, p. 184.

37. LIGHTHILL, M. J.: A Technique for Rendering Approximate Solutions toPhysical Problems Uniformly Valid. Phil. Mag., Ser. 7, vol. 40, no. 311,Dec. 1949, pp. 1179-1201.

38. KEVORKIAN, J.: The Two Variable Expansion Procedure for the Approxi-mate Solution of Certain Non-Linear Differential Equations. Rep.SM-42620, Douglas Aircraft Co., Inc. (AD-437675), Dec. 3, 1962.

39. KOPAL, ZDEAK; Numerical Analysis. John Wiley & Sons, Inc., 1955.40. MILNE, WILLIAM E.: Numerical Solution of Differential Equations. John

Wiley & Sons, Inc., 1953.41. COLLATZ, L.: The Numerical Treatment of Differential Equations. Springer-

Verlag, 1960.42. Fox, L.: The Numerical Solution of Two-Point Boundary Problems in

Ordinary Differential Equations. Clarendon Press, Oxford, 1957.43. Fox, LESLIE; and MAYERS, D. F.: Computing Methods for Scientists and

Engineers. Clarendon Press, Oxford, 1968.44. HILDEBRAND, F. B.: Introduction to Numerical Analysis. McGraw-Hill

Book Co., Inc., 1956.45. 'C,ONTE, S. D.: Elementary Numerical Analysis -An Algorithmic Approach.

McGraw Hill nook Co Inc 1965.46. RALSTON, ANTHONY; and WILF, HERBERT S., eds.: Mathematical Methods

for Digital Computers. Vol. I. John Wiley & Sons, Inc., 1960.47. VARGA, RICHARD S.: Matrix Iterative Analysis, Prentice-Hall, Inc., 1962.48. BELLMAN, R.; and KALABA, R.: Quasilinearization and Nonlinear Boundary

Valve Problems. American Elsevier, 1965.

349/350

Absolute value of complev number. 113Adams' method, 319Adams-Bashforth formula, 340Adams-Moulton formula, 340Adjoint equation, 99, 105Affine group, 93Airy's equation. 288Analytic continuation

along a simple curve. 134defitition, 133in neighborhood of singular points. 137of multiple-valued functions, 139of solution of differential equation, 160specific method, 135

Analytic functions of a complex variable, 114, 116Analytic functions of is complex variables, 146Associated homogeneous equation, 24Associated Legendre equation. 229Associated Legendre functions, 231Asymptotic expansion, 262, 265

nonuniformly valid, 287of a solution. 267. 285

Asymptotic power series, 265Asymptotic sequence of functions. 264Asymptotic series, defined. 263, 265Autonomous systems of differential equations, 59

Bernoulli's equation. 45Bessel's equation. 241. 277, 288Besse' function, 278

of first kind. 242of second kind, 244modified, of first and third kind, 246

Beta function, 156Boundary conditions, 17, 319,341Branch cut, 144Branch of analytic function, 145Branch point, 141

Canonical basis of solutions, 168Cauchy estimates, 123

INDEX

Cauchy product of two series, 126Cauchy-Riemann equations, 114Characteristic curves of partial differential equation, 68Characteristic equation

of partial differential equation, 68of two algebraic equations, 162

Characteristic exponents, 177difference an integer, 195difference not an integer, 188

Circle of convergence, 122Complementary function, 32Complete analytic function, 139Complex integration, 129Complex plane, 111, 116Complex variables, IllConfluence of singularities in Riemann-Papperitz

equation, 233Confluent hypergeometric functions, 235Conformal mapping, 116Critical points of first-order autonomous systems, 64

Degree of differential equation. 8Derivatives

ot !unctions at complex variables, 113of vectors, defined, 58

Differenceforward, backward, central, 334

Difference equations, 320one-step, 322n-step, 334, 336

Differential equationcontaining a parameter, 281definition, 7exact, second order, 93equivalent to linear equation, 101fundamental theorem, 13, 148, 151homogeneous, 8linear, 8normal, 13of first degree, 35of first order, 35partial, 68, 6;

351


Differential equation in complex plane, 148analytic continuation of solution, 150Taylor series expansion of solution. 149

Discretization error (see truncation error)Divergent series, as solution of differential equations. 256Domain, 2, 114, 121, 123

Elliptic equation, 79Elliptic functions, 79Entire function, 121Equation of Painleve. 102Equation splitting, 109Equation with dependent variable missing, 74Equilibrium solution of first-order autonomous system, 64Equivalent equations under change of variable (second

order), 105Essential singularity, 120

of p(z) and q(z), 252Euler's constant, 245

equation, 83

equation. homogeneous. 107formula. 112integral of second kind, 154method, 319, 324method improved, 339

Factorial function, generalized. 155Finite-difference method, 341Finite points of the complex plane, 117, 121First integral, 12Fuchs theorem, 149Fuchsian equation, 201

with two singular points, 202with three singular points. 203

Fuchsian invariant for equation of order two. 206Function

implicit, 5single valued, 3

Functionally independent integrals of a system, 55Fundamental set of equations. 151, 159, 168Fundamental solutions, 33Fundamental theorem

of analytic continuation, 132, 141of first-order normal system, 50

Gamma function, 154Gegenbauer polynomial, 227G....metric series, 125Group theory method for reducing order of equation, 91

Hankel functions of first and second kind, 246Hermite polynomial, 249Holomorphic functions of a complex variable, 114

352

Homogeneous equationfirst order, 43of order n, 8second order, 79

Homogeneous Euler equation, 107Homogeneous function of degree k, 45Homogeneous solution, 24Hypergeometric equation 210

derivative of, 223integral form. 222Kummer's 24 solutions, 220logarithmic solution. 225of Gauss. 212Whittaker's form, 238

Hypergeometric functions. 215generalized, 224

Implicit function theorem, 5Indicial equation, 177, 199

of Fuchsian equation with two singular points, 203of Fuchsian equation with three singular points, 205

Infinitesimal operator, 87Infinitesimal transformations, 87Initial zonditions, 13, 319Inner and outer expansions of solution near a transition

point. 297Instability

in predictor - corrector methods, 340induced, 322inherent, 321

Integralfirst, 95of diffprPntial ecloAtinna, 12of system of equations, 51

Integral curves of first-order autonomous systems, 64Integrating factor

of first-order equation, 41of second-order equation, 97

Interchange of dependent and independent variables, 107Invariant of second-order equation, 105Irregular singular points, 173, 251Isobaric equation, 43, 81, 93Isobaric function, 82Isolated singular points

of analytic functions, 119of second-order linear equation. 159

Jacobi differential equation, 225Jacobi polynomial, 226Jacobian determinant, 52

Kummer's confluent hypergeometric equation, 235first formula, 240

second formula, 24324 solutions to hypergeometric equation, 220

Laguerre polynomials, 240Lame equation, 249Laurent series, 128Legendre

equation, 228function of first kind, 228function of second kind, 228polyroimial, 228transform. 108

Linear equationsdifferential. 24first order, 35

Linear fractional transformation. 117, 209Linear homogeneous equation, 106Linear independence, 25Linear second-order equations. 74Linearly independent solutions in complex plane. 151Liouville's theorem. 121Logarithmic function of complex variable, 145

Magnification group. 88Matched asymptotic expansions, 295Mathieu equation, 249Meromorphic function, 121Method of variation of parameters, 84Milne's method, 340Modulus of complex number, 113Monodromy theorem. 141 (footnote)Multiple-time methods, 311Multiple-valued function, 139

n-dimensional vectors, 57Neighborhood. 2Nonuniform-distortion group. 93Normal differential wruation, 12Normal form of second-order equation. 105Normal solutions of irregular singular points. 269Normal systems of equations. 49

One-dimensional equation, 43One-step processes, 322Order

defined, 263of a differential equation, 8symbols, 263

Ordinary point, 159

Parabolic cylinder functions, 248Parameter, expansions of solutions in, 282Partition of an interval, 320p-discriminant, 23Pockhammer-Barnes function, 236

INDEX

Point at infinity. 117. 173Polar representation of complex number, 112Pole of an analytic function, 119Polynomial, 121Power series in complex variable, 122Predictor-corrector methods, 337Primitive, 9Principal branch of the logarithm. 145Principle of permanence of functional relations, 146

Quasi-linear partial differential equation, 69Quasi-linearization of nonlinear equations, 344

Radius of convergence, 122, 160Rank of an equation, 269Rational function. 121Recurrence relation, 177, 188, 190, 196Regular singular point, 172Regular solutions, 170Removable singularity, 119Riccati equation. 46, 80Riemann-Papperitz equation, 205Riemann P-symbol, 206Rodrigues formula for Jacobi polynomials, 226Round-off error, 321Runge-Kutta method, 328, 332

Second-order equations, 73with dependent variable missing, 74with independent variable missing, 77

Secular terms of a solution, 307Separable equation (first order), 43Shooting methods. 343Simple pole, 119Simply connected domain, 38Single-parameter Lie group, 86Singular locus, 23Singular point, 25

of a differential equation, 35, 159Singular solutions, 19. 21Singularity of solutions, 169Solution

curves, 64of differential equation, 8of equation, 5general, 10particular, 8singular. 11. 19

Solution of second-order linear equationat ordinary point, 174at regular singular point, 174by change of variables, 104near imgular singular point, 251near isolated singular point, 160

Stokes phenomenon, 266. 277

353


Strained coordinates, method of, 306Subnormal solutions of irregular singular points, 274Superposition principle, 24Systems of equations, normal, 49

Tables of differential equations and solutions, 110Taylor series, 123Taylor series method, 322Transformation to equation with constant coefficients, 106Transition point, 287Translation group, 92Truncation error, 320

local, 322 .

354

Tschebychev equation, 228Tschebychev polynomial, 228

Ultraspherical equation, 227Ultraspherical polynomial, 227 (footnote)

Vector functions, 58

Weber equation, 248Weber function, 248Whittaker's confluent hypergeometric equation, 238Wronskian, 103, 152, 170

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