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Appendix B
ORDINARY DIFFERENTIAL EQUATIONS AND SPECIAL FUNCTIONS
B.1 INTRODUCTION
This appendix summarizes methods for solving the types of ordinary differential
equations which are encountered most frequently in transport problems. In some cases the
solutions involve special functions, whose properties are also discussed. It is assumed that the
reader is already familiar with these methods, and what is presented is intended only as a concise
review. Additional explanation may be found in Hildebrand (1976) or in any introductory text
on differential equations, such as Rabenstein (1966). Abramowitz and Stegun (1970) is an
authoritative and comprehensive source for the properties of special functions.
In each equation the unknown function is denoted as y( x). In general, the first
consideration is whether the equation is linear or nonlinear . Except for separable first-order
equations, only linear differential equations are discussed. For an equation to be linear, the
coefficients of y and its derivatives all must be independent of y, and there can be no nonlinear
functions of y such as e y. Except for first-order separable equations, and ones which contain a
small parameter and therefore permit use of perturbation methods (see Chapter 4), there are few
ways to find analytical solutions to nonlinear problems; numerical methods are usually needed.
If the differential equation is linear, the next consideration is whether the coefficients of y
and its derivatives are constants or functions of x. If each is constant, the solution procedure isstraightforward; if one or more depends on x, success or failure in obtaining a useful analytical
result may hinge on the extent to which that equation has been studied previously, and its
solutions documented. The defining feature of many special functions (such as Bessel functions)
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is that they are solutions to certain differential equations. Making good use of available
knowledge requires familiarity with the differential equations that give rise to well-known
special functions. Several such second-order equations will be discussed.
Whether the coefficients are constant or not, a differential equation is either
homogeneous or nonhomogeneous . In a homogeneous equation y( x) appears in each term, and a
hallmark of such equations is that one possible solution is y = 0. The general solution of an nth-
order, linear homogeneous equation is a sum of n fundamental solutions. Each fundamental
solution is weighted by a constant, and the n constants are evaluated by applying the boundary
conditions. Transport models usually involve boundary-value problems (where boundary
conditions are imposed at two locations) rather than initial-value problems (where all
information is at one position, typically x = 0). If the equation is nonhomogeneous, the general
solution is the homogeneous part plus a particular solution (i.e., any solution to the full equation,
boundary conditions aside). The particular solution is chosen only to satisfy the differential
equation; the n constants in the homogeneous solution are determined still by the boundary
conditions.
First-order differential equations (separable or linear) are discussed in Section B.2 and
nth-order equations with constant coefficients are reviewed in Section B.3. Fundamental
solutions for equations with constant coefficients are tabulated for convenient reference, as are
common forms of particular solutions. The remainder of this appendix concerns linear
differential equations with variable coefficients, and the corresponding special functions. Theequations selected are ones which arise repeatedly in this book. The differential equations that
yield cylindrical or spherical Bessel functions, and the properties of those functions, are the
subject of Section B.4. Certain other equations with variable coefficients are discussed in
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Section B.5. Except for the nth-order equidimensional equation (Section B.5), those with
variable coefficients are all second order, and only their homogeneous forms are discussed. A
few less-common differential equations are mentioned as they appear in various problems and
are not included here. If an equation of interest is not found, it may be worthwhile to consult the
extensive compilation of solutions in Kamke (1943).
B.2 FIRST-ORDER EQUATIONS
Separable
A separable first-order equation has the form
dydx
=
f ( x) g ( y)
. (B.2-1)
Unless g is a constant, this differential equation will be nonlinear. However, it can always be
integrated as
g dy = f dx . (B.2-2)
Whether the resulting solution is implicit or explicit depends on g( y). If g( y) = yb with b > 0, the
result is
y( x) = (b + 1) f dx 1/ ( b+ 1)
+ C (B.2-3)
where C is a constant.
Linear
Linear first-order equations are of the form
dy
dx+ a1 ( x) y = h( x) . (B.2-4)
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Such equations are exceptional in that no special procedure is needed if they are
nonhomogeneous (i.e., if h 0). With the function p( x) evaluated as
p( x) = exp a1 ( x) dx (B.2-5)
the general solution is
y( x) =C
p( x)+
1
p( x) p( x)h( x) dx (B.2-6)
where C again is a constant.
B.3 EQUATIONS WITH CONSTANT COEFFICIENTS
A linear, nth-order equation with constant coefficients can always be written as
d n y
dx n+ a1
d n1 y
dx n1
+ ... + a n 1dy
dx+ a n y = h( x) . (B.3-1)
If h( x) = 0, the equation is homogeneous. Associated with Eq. (B.3-1) is a characteristic
equation, the n roots of which determine the general solution to the homogeneous differential
equation. The characteristic equation, which is obtained by inserting erx into Eq. (B.3-1), is
r n
+ a 1 r n 1
+ ... + an 1 r + a n = 0 . (B.3-2)
The solutions which correspond to different types of roots are summarized in Table B-1, in
which C i and D i are constants. It is assumed here that the coefficients a i are all real, in which
case there are always n real solutions. Repeated or complex conjugate roots each yield more
than one fundamental solution, as shown. For r = 1, as with d 2 y/dx2 - y = 0, the two
fundamental solutions can be written either as ( e x, e- x) or (sinh x, cosh x). The exponentials tend to
be more convenient for infinite or semi-infinite domains, and the hyperbolic functions better for
finite domains.
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Table B-1. General Solutions for Homogeneous Differential Equations with Constant
Coefficients
Root of Characteristic Equation Homogeneous Solution
r a single root (real) Ce rx (A)
r an m-fold root (real) e rx C 0
+ C 1 x + ... + C
m 1 xm 1( ) (B)
r = a bi (complex, each a single root) e ax C cos bx + D sin bx( ) (C)
r = a bi (complex, each an m-fold root) e ax cos bx C 0
+ C 1 x + ... + C
m 1 xm 1( )
+ e ax sin bx D0
+ D1 x + ... + D
m 1 xm 1( ) (D)
If the differential equation is nonhomogeneous, a particular solution must be added to the
homogeneous solution. If h( x) happens to be a solution of some linear, homogeneous differential
equation with constant coefficients, then the method of undetermined coefficients can be used to
find the particular solution, as summarized below. If not, a more general but usually lengthier
procedure called variation of parameters can be used. Variation of parameters will yield the
particular solution for any linear equation (Hildebrand, 1976; Rabenstein, 1966).
Particular solutions corresponding to various functions h( x) are shown in Table B-2.
After substituting the trial particular solution into the differential equation, the constants are
chosen so that the nonhomogeneous form of Eq. (B.3-1) is satisfied. If any term in the given
form of the particular solution appears also in the homogeneous solution, the entire particular
solution must be multiplied by xk , where k is the smallest positive integer that prevents the
duplication. If h( x) consists of a sum of terms, the solutions corresponding to each may be added
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to find the complete particular solution. If h( x) = c, a constant, then the particular solution is
simply y = c/a n.
Table B-2. Particular Solutions for Nonhomogeneous Differential Equations with Constant
Coefficients
Nonhomogeneous Term, h( x) Particular Solution
Cx m A 0 + A1 x + ... + A m xm (A)
Cx m e ax A 0 + A1 x + ... + A m xm( )e ax (B)
Cx m e ax cos bx or Cx m e ax sin bx A 0 + A1 x + ... + A m xm( )e ax cos bx
+ B0
+ B1 x + ... + B
m x m( )e ax sin bx
(C)
B.4 BESSEL AND SPHERICAL BESSEL EQUATIONS
Bessel Functions
The general form of Bessels equation is
xd dx
xdydx
+ m2 x 2 v2( ) y = 0 (B.4-1)
where m is a parameter and is any real constant. The form usually encountered, as with
conduction or diffusion problems in cylindrical coordinates, has = 0. Setting = 0 and dividing
by x2 gives
1
xd dx
xdydx
+ m2 y = 0 . (B.4-2)
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The solutions of Bessels equation have been studied extensively (Watson, 1944). The two
linearly independent solutions to Eq. (B.4-1) are written as J (mx) and Y (mx), and are known as
Bessel functions of order of the first and second kind, respectively. The solutions to Eq. (B.4-
2) are Bessel functions of order zero, J 0(mx) and Y 0(mx). Bessel functions of integer order are
widely available in spreadsheet programs and other software for personal computers, making
calculations with them routine.
As discussed shortly, the derivatives and integrals of J 0 and Y 0 can each be expressed in
terms of the corresponding Bessel functions of order one. Accordingly, familiarity with the
properties of J 0, J 1, Y 0, and Y 1 is sufficient for the problems encountered in this book. Graphs of these functions are shown in Fig. B-1. All four functions are oscillatory, although with variable
periods and amplitudes, and have infinitely many roots. Two values worth noting are J 0(0) = 1
and J 1(0) = 0. An important distinction between Bessel functions of the first and second kinds is
that J 0(0) and J 1(0) are finite, whereas Y 0(0) and Y 1(0) are not.
Numerous Bessel-function identities may be found in Watson (1944) and Abramowitz
and Stegun (1970). Ones which are helpful in evaluating derivatives and integrals are
dJ 0 (mx )dx
= mJ 1(mx ) ,
d
dx xJ
1(mx ) = mxJ 0 (mx ) (B.4-3a,b)
dY 0 (mx )dx
= mY 1(mx ) ,
d
dx xY
1(mx ) = mxY 0 (mx ) . (B.4-4a,b)
Using Eq. (B.4-3b), the integral of xJ 0 over the interval [0, L] is
J 0 (mx ) x dx
0
L
= L
m J
1(mL ) (B.4-5)
which is useful in constructing Fourier-Bessel series (Chapter 5). Also needed for such series is
the definite integral of xJ 02, which is given by
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J 02 (mx ) x dx
0
L
=m 2
2 J
02 (mL ) + J 1
2 (mL ) . (B.4-6)
This last identity can be derived from Eqs. (B.4-2) and (B.4-3a), as detailed in Section 4.7 of the
first edition of this book.
Modified Bessel Functions
A differential equation closely related to Bessels equation, but with very different solutions, is
the modified Bessels equation . Its general form is
x d dx
x dydx m
2 x 2 + v2( ) y = 0 (B.4-7)
where again m is a parameter and is any real constant. Equations (B.4-1) and (B.4-7) differ
only in the sign of the m2 x2 term. The solutions of Eq. (B.4-7) are written as I (mx) and K (mx),
and are called modified Bessel functions of order of the first and second kind, respectively. As
with Bessel functions, software for computing modified Bessel functions of integer order is
widely available. The differential equation with = 0 again is the one of greatest interest. For
that case Eq. (B.4-7) can be rewritten as
1
xd dx
xdydx
m2 y = 0 . (B.4-8)
Modified Bessel functions of orders zero and one are plotted in Fig. B-2. The most
obvious difference between Bessel functions and modified Bessel functions is that the latter do
not oscillate or have multiple roots. The limiting values of the modified Bessel functions are
I 0 (0) = 1 , I 1(0) = 0 , I 0 ( ) = , I 1( ) = (B.4-9)
K 0 (0) = , K 1(0) = , K 0 ( ) = 0 , K 1( ) = 0 . (B.4-10)
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Identities which are helpful in evaluating derivatives and integrals are
dI 0 (mx )dx
= mI 1(mx ) ,
d
dx xI
1(mx ) = mxI
0 (mx ) (B.4-11a,b)
dK 0 (mx )dx
= mK 1(mx ) ,
d
dx xK
1(mx ) = mxK 0 (mx ) . (B.4-12a,b)
As with Bessel functions, differentiation or integration of modified Bessel functions of order
zero requires knowledge only of the corresponding functions of order one.
Spherical Bessel Functions
The spherical Bessels equation is written generally as
d dx
x2dydx
+ m2 x2 n(n + 1) y = 0 (B.4-13)
where m is any real constant and n is a non-negative integer. The solutions, called spherical
Bessel functions, may be expressed in terms of Bessel functions of order n + (1/2). Accordingly,
their properties are covered in discussions of Bessel functions of fractional order (Watson, 1944;
Abramowitz and Stegun, 1970). The form of Eq. (B.4-13) encountered in conduction or
diffusion problems in spherical coordinates is that with n = 0, or
1
x 2d dx
x2dydx
+ m2 y = 0 . (B.4-14)
In this case no special functions are required. The general solution of Eq. (B.4-14) is
y(mx) = Asin mx
mx+ B
cos mxmx
(B.4-15)
where A and B are constants. In that they are oscillatory functions with infinitely many roots, the
fundamental solutions in Eq. (B.4-15) are somewhat akin to the Bessel functions J 0 and Y 0. In
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this case the periods are constant and only the amplitudes vary. Also, as with Y 0, one of the
solutions is unbounded at x = 0. That is, for x 0, sin( mx)/mx 1 but cos( mx)/mx . The
spherical Bessel functions of order zero are plotted in Fig. B-3.
Modified Spherical Bessel Functions
A differential equation closely related to Eq. (B.4-13) is the modified spherical Bessels
equation,
d dx
x 2dydx
m2 x 2 + n n + 1( ) y = 0 . (B.4-16)
As with the corresponding equations for cylindrical problems, Eqs. (B.4-13) and (B.4-16) differ
only in the sign of the m2 x2 term. The solutions to Eq. (B.4-16), called modified spherical Bessel
functions, are expressible in terms of modified Bessel functions of order n + (1/2). The most
commonly encountered form of the differential equation is that with n = 0, in which case
1
x2
d
dx x2
dy
dx
m
2 y = 0 . (B.4-17)
The general solution to this equation is expressible in terms of elementary functions as
y(mx) = Asinh mx
mx+ B
cosh mxmx
= C emx
mx+ D
emx
mx. (B.4-18)
When using the hyperbolic form of Eq. (B.4-18), both solutions are unbounded at x = but one
is finite at x = 0. That is, for x 0, sinh( mx)/mx 1 but cosh( mx)/mx . The hyperbolic
forms of the modified spherical Bessel functions of order zero are shown in Fig. B-3. When
using the exponential form of Eq. (B.4-18), both solutions are unbounded at x = 0 but one is
finite at x = . Accordingly, the hyperbolic solutions are best for finite domains that include x =
0, and the exponential ones are preferred for semi-infinite domains that exclude the origin.
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The correspondence between the solutions of the ordinary or modified spherical Bessels
equations and those of the analogous equations in Cartesian coordinates ( d 2 y/dx2 m2 y = 0) is
noteworthy. In each case the spherical solution is the Cartesian one divided by x. This underlies
a transformation that is sometimes used in solving spherical conduction or diffusion problems, in
which there is a change in the dependent variable given by (r,t ) = (r,t )/r . This transforms a
problem for (r,t ) involving the spherical 2 operator into a problem for (r,t ) involving the
Cartesian one.
Also needed sometimes is the modified spherical Bessels equation with n = 1, which is
d dx
x 2dydx
m2 x2 + 2 y = 0 (B.4-19)
This is encountered, for example, in certain spherical problems involving diffusion with first-
order reactions. Again, the solutions can be expressed in terms of elementary functions. The
general solution is
y(mx)=
A sinh mx(mx)2
+cosh mx
mx
+
B
sinh mxmx
cosh mx(mx)2
. (B.4-20)
Neither of the fundamental solutions is finite at x = 0 or x = . In unbounded domains, the
exponential form of the general solution is preferable, which is
y(mx) = C emx
mx1 1
mx
+ Demx
mx1 +
1mx
. (B.4-21)
Although both solutions are still unbounded at x = 0, one remains finite now at x =
.
B.5 OTHER EQUATIONS WITH VARIABLE COEFFICIENTS
Equidimensional Equations
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An nth-order equidimensional equation (also called an Euler equation or Cauchy-type equation )
has the form
xnd n y
dx n+ b1 x
n 1 d n 1 y
dx n
1
+ ... + bn 1 xdy
dx+ bn y = h( x) . (B.5-1)
The characteristic equation is
r (r 1)...( r n + 1)[ ]+ b1 r (r 1)...( r n + 2)[ ]+ ... + b n 1r + b n = 0 . (B.5-2)
In the simplest situations (single roots, all real), the homogeneous solutions are of the form Cx r.
The solutions for various types of roots are summarized in Table B-3. Particular solutions for
certain nonhomogeneous equidimensional equations are given in Table B-4. In the last entry, k
is the smallest positive integer that will prevent the particular solution from duplicating any part
of the homogeneous solution. As with any other linear differential equation, if h( x) consists of a
sum of terms, the solutions corresponding to each may be added to find the complete particular
solution. Also, if h( x) = c (a constant), the particular solution is just y = c/bn.
Table B-3. General Solutions for Homogeneous Equidimensional Equations
Root of Characteristic Equation Homogeneous Solution
r a single root (real) Cx r (A)
r an m-fold root (real) x r C 0
+ C 1
ln x + ... + C m 1 ln x( )
m 1
(B)
r = a bi (complex, each a single root) x a C cos( b ln x ) + D sin( b ln x )[ ] (C)
r = a bi (complex, each an m-fold root) x a cos( b ln x ) C 0 + C 1 ln x + ... + C m 1 (ln x )m 1
+ x a sin( b ln x ) D 0 + D 1 ln x + ... + D m 1 (ln x )m 1
(D)
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Table B-4. Particular Solutions for Nonhomogeneous Equidimensional Equations
Nonhomogeneous Term, h(x) Particular Solution
xs (s r) Ax s (A)
xs (s = r) Ax s (ln x )k (B)
Error Function
A differential equation which arises in similarity solutions to transient diffusion or conduction
problems (Chapter 4) is
d 2 y
dx 2+ 2 x
dydx
= 0 . (B.5-3)
This is equivalent to a first-order linear equation governing the function dy/dx , so that dy/dx is
found as in Section B.1. Another integration gives the general solution as
y( x) = a e x2
dx + b (B.5-4)
where a and b are constants.
To obtain a form that is more convenient computationally, Eq. (B.5-4) is rewritten using
a definite integral as
y( x) y(0) = a e s2
ds0
x
(B.5-5)
where y(0) takes the place of b. The error function , which arises in probability theory and is
widely available in commercial software, is defined as
erf( x ) 2
e s2
ds
0
x
. (B.5-6)
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Because erf( x) contains the same definite integral as Eq. (B.5-5), the solution to Eq. (B.5-3) can
be rewritten as
y( x) = Aerf( x) + B (B.5-7)
where A and B are constants.
The complementary error function is
erfc( x ) 2
e s2
ds x
= 1 erf( x ) . (B.5-8)
With erfc( x) being linearly related to erf( x). the general solution to Eq. (B.5-3) can be written
also as
y( x) = C erfc( x) + D (B.5-9)
where C and D are constants.
Whether erf or erfc is more convenient for a particular problem will depend on the
boundary conditions. The limiting values (for positive arguments) are
erf(0) = 0 , erf( ) = 1 , erfc(0) = 1 , erfc( ) = 0 . (B.5-10)
The error function and complementary error function are plotted in Fig. B-4. Although not
shown, erf may also have negative arguments; it is an odd function [erf(- x) = -erf( x)].
Gamma and Incomplete Gamma Functions
A more general version of Eq. (B.5-3) is
d 2
ydx 2
+ nx n 1 dydx
= 0 (B.5-11)
where n is any positive integer. Following the same reasoning as with error functions, the
solution may be written as a definite integral of exp(- xn). That integral can be evaluated using
incomplete gamma functions, as will be described.
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The gamma function , ( z ), is defined generally as
( z) t z1e t dt 0
(B.5-12)
where z may be complex (Abramowitz and Stegun, 1970, p. 255). However, specializing by
making the substitutions t = sn and z = 1/ n, where n is a positive integer, it is found that
e sn
ds
0
= (1 / n )
n. (B.5-13)
Four values are (1) = 1, (1/2) = , (1/3) = 2.67894, and (1/4) = 3.62560. If the
integration is terminated at s = x, the result is an incomplete gamma function. For fractional
arguments, the incomplete gamma function has normalized and non-normalized forms denoted
as P (1/n, x) and (1/n, x), respectively. They are related to the integral in Eq. (B.5-13) as
P (1 / n , x ) =n
(1 / n )e s
n
ds
0
x
= (1 /n , x )
(1 / n ). (B.5-14)
Thus, P varies from 0 to 1 as x goes from 0 to , similar to the error function. Indeed, P (1/2, x)
= erf( x).
It follows that the general solution of Eq. (B.5-11) may be written as
y( x) = AP (1 / n, x) + B (B.5-15)
which is analogous to Eq. (B.5-7). As with the error function, incomplete gamma functions have
applications in probability theory. However, software for them is less widely available.
The definition in Eq. (B.5-12) indicates that incomplete gamma functions may be used to
evaluate a much broader class of definite integrals. For example, setting z = k/n gives
P (k / n , x ) =n
(k / n )s k 1e s
n
ds
0
x
= (k / n , x )
(k / n ). (B.5-16)
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Legendre Polynomials
Certain conduction or diffusion problems in spherical coordinates lead to Legendres equation ,
d dx
1 x2( )dydx
+ (n + 1) ny = 0 (B.5-17)
where n is a non-negative integer. The solutions of Eq. (B.5-17) are detailed in Hobson (1955).
In the usual applications of Legendres equation the interval for x is [-1,1], and it is found that
nontrivial solutions which are bounded at x = 1 exist only if n is as stated above. Moreover,
there is only one such bounded solution for a given value of n. It is
y( x) = AP n ( x) (B.5-18)
where A is a constant and the functions P n( x) are Legendre polynomials. (The other linearly
independent solution of Eq. (B.5-17), which is unbounded at x = 1 and therefore not of interest
here, involves what are called Legendre polynomials of the second kind.)
The first two Legendre polynomials are P 0( x) = 1 and P 1( x) = x, and the remainder can be
generated using the recursion relation,
P n + 1( x ) =
(2 n + 1) xP n ( x ) nP n 1( x )n + 1
. (B.5-19)
Alternatively, they can be computed using Rodrigues formula,
P n ( x ) =
1
2 n n !
d n
dx n( x 2 1) n . (B.5-20)
The first six Legendre polynomials are given in Table B-5. They are standardized such that
P n(1) = 1. The functions with even and odd values of n contain only even and odd powers of x,
respectively. Accordingly, even-numbered Legendre polynomials are even functions [ P n( x) =
P n(- x)] and odd-numbered ones are odd functions [ P n( x) = - P n(- x)].
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Table B-5. Legendre Polynomials
n 0 1 2 3 4 5
P n( x) 1 x 12
3 x 2 1( )
1
25 x
3 3 x ( ) 18
35 x 4 30 x 2 + 3( ) 18 63 x5 70 x 3 + 15 x( )
References
Abramowitz, M. and I. A. Stegun. Handbook of Mathematical Functions. U.S. Department of
Commerce, National Bureau of Standards, Washington, DC, 1970.
Hildebrand, F.B., Advanced Calculus for Applications , Second Edition. Prentice-Hall,
Englewood Cliffs, NJ, 1976.
Hobson, E. W. The Theory of Spherical and Ellipsoidal Harmonics . Chelsea, New York, 1955.
Kamke, E. Differentialgleichungen, Vol. 1. Akademische Verlagsgesellschaft Becker & Erler
Kom.-Ges., Leipzig, Germany, 1943.
Rabenstein, A. L. Introduction to Ordinary Differential Equations . Academic Press, New York,
1966.
Watson, G. N. A Treatise on the Theory of Bessel Functions, Second Edition. Cambridge
University Press, London, 1944.
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-1
-0.5
0
0.5
1
0 2 4 6 8 10 12x
J0(x)
J1(x)
Y0(x)
Y1(x)
Figure B-1. Bessel functions of orders 0 and 1.
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0
1
2
3
0 1 2 3x
I0(x)
I1(x)
K1(x)
K0(x)
Figure B-2. Modified Bessel functions of orders 0 and 1.
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-0.5
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5
x
cosh(x)/x
sinh(x)/x
sin(x)/x
cos(x)/x
Figure B-3. Spherical Bessel functions and modified spherical Bessel functions of order 0.
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0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
x
erf(x)
erfc(x)
Figure B-4. Error function and complementary error function.
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