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MATHEMATICAL SUPPLEMENT: PART ITAYLOR S SERIES

T H E EXPANSION of functions into series representations is a C01111110nly used an deffective analytical technique. In electromagnetic theory th e function to be expandedoften depends on several variables, an d thus it is desi rabl e to deve lop such a techniquewith adequate generality. Accordingly this small supplement on series, after a briefhistorical introduction, reviews several mean value theorems, derives Taylor s series forfunctions of one variable, a nd t he n extends th e result to cover multivariable functions.

s.i * HISTORICAL SURVEY

The series expansionh 2

f( x + h) f(x) + hf'(x) + 2 f (x) +which bea rs his name was first enunciated by Brook Taylor (1685-1731) as early as1712 in a letter to John Machin, Its first formal appearance was in his text 1v[ethodusincrementorum directaet inversa which was published in London in th e period 1715-1717.Th is t ex t also contains th e easy consequence no w known as Maclaurin s series, butTaylor s proof of these expansions did no t consider convergence an d is worthless. Theimportance of these expansions was no t appreciated by analysts for over a half centuryuntil Lagrange point ed out their applicability, an d no rigorous proof of Taylor stheorem was offered until Cauchy included a remainder term and tes ted for convergencein 1821.

Colin Maclaurin (1698-1746), though an able mathematician, is improperly credited

with authorship of th e expansionx 2

f(x) f(O) + xf'(O) + , f ( O ) + 2.

which was contained in his Treatise of Fluxions published in Edinburgh in 1742. Thisexpansion is obviously a spec ial case of Taylor s theorem, a point w hi ch w as indicatedby Taylor 25 years earlier. Additionally, Maclaurin s expansion was apparently discovered independently by James Stirling an d is contained in his paper Methodusdifferentialis sive Tractaius de summoiione et interpolatione serierum infiniiarum publ ished in London in 1730. The greater f ame of Maclaurin and th e wider circulation ofhis Treatise are accountable for this miscredit.

*This section ma y be omitted without loss in continuity of th e technical presentation.

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558 Taylor's Series

5.2 MEAN VALUE THEOREMS

IVlATHEl\IATICAL SUPPLEl\1ENT: PAR r I

A discussion of Taylor s series builds on th e base of several mean value theorems which

serve as lemmas. The first of these is th e well-known

ROLLE S THEOREl vI: Let f(x) be a function of the real »ariable x which possesses a continuous first derivative over the interval Xl X X2. Let a and b be two points within theintervalt for which f(a) = f(b) O Then at least one value of x can befound between a andb, say x == t, for which f (t) O

Proof: The truth of this theorem is almost self-evident from a geometric display of th e

function such as shown in Figure S.l. If th e function is to be zero a t a an d a t b it cannot

f(x)

J - - - - - - ; - - - 4 - - - - - - - - 4 L . - - - - - ~ - o _ _ ~ _ _ _ - - x

FIGURE S.l Rolle's theorem.

be ever-increasing, no r ca n i t be ever-decreasing in the interval between a an d b. Wherethe function changes over from increasing to decreasing, th e first derivative must vanish.

Rol le s theorem can be em ployed to establish th e

THEOREIVI OF l'vIEAN VAL D E: Let f(x) and g(x) be two functions of the real variable x which

possess continuous first derivatives ihrouqhoui the intervalXl

SX X2.

Let a and b beanytwo points within this in ierool such that g(a) g(b). I f g' (z) is nowhere zero in the interval,then for some value x t between a and b,

f(b) - f(a) _ f ( ~ )

g(b) - g(a) - g ( ~ )

Proof: Define a function h(x) by th e relation

(S.l)

hex) = ~ ~ ~= [g(x) - g(a)] - [f(x) - f(a)]

t In this an d all subsequent theorems of this supplement, b can be either larger or smaller than a.

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SECTION 8.2 Mean Value Theorems 559

I t can be observed that hex) is a function which satisfies all the requirements of Rolle s

theorem. I t ha s a continuous first derivative in th e interval an d h(a) h(b) = O 8ince

h'(x) = feb) - f(a) g '(x) - f'(x)g(b) - g(a)

it follows that for some x ~ ,

h'(O = 0 = feb) - f(a) g m - 1 (0g(b) - g(a)

which, upon rearrangement, yields t he s ta te d result.A special case of t hi s t he or em of some importance occur s when g(x) = x. Then

Equation (S.l) reduces to

feb) - ita) = I'mb - a

A significant generalization of t he a bo ve t he or em is embodied in th e

(8.2)

EXTENDED THEOHElVI OF ~ I E A NVALUE: Let f(x) be any function of the real variable xwhich, together with its first n derivatives, is coniinuous in the interval X l :: ; X :: ; X2. Lei aand b be any two points within this interval. Then

b b - a ) 2f(b) f(a) + - , - f a ) + ) f (a) +

1. ..,.

in which ~ n is some point between a and b.

Proof: If one makes use of Equation (8.2), there is a point ~ obetween a and b for which

feb) - f(a) - (b - , a) f ~ o )= 01.

Define a constant K 2 by th e equation

feb) - f(a) - (b - a) rea) _ (b - a)2 K2

= 0I 2

an d from this form th e function

(x - a) (x - a)2

h(x) ==f(x) - f a _ 1 f'(a) - 2 K 2

The function hex) ha s a continuous first derivative in th e interval, g iven by

h'(x) f'ex) - f'ea) - (x - a)K 2

(8.4)

and since h(a) == h(b) 0, Rolle s theorem applies. Thus there is a point x == ~ 1between

a an d b such that h' ( ~ 1 ) O.Furthermore, h (x) ha s a continuous first derivative in th e interval , namely

h (x) == f (x) - K 2

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560 Taylor's Series IVIATHElVrATICAL SUPPLEMENT: PART I

an d since h (a) h ( ~ l ) 0, there must be a point x ~ 2 between a an d ~ 1for which

If t hi s r esul t is substituted in (8.4), one obtains

f(b) = f(a) + (b - a) f'(a) + (b - a ) 2 f ( ~ )1 2 2

A constant K 3 can n ext be defined by th e relation

(8.5)

f(b) - f(a) - (b - a) f'(a) _ (b - a)2f (a) _ (b - a)3 K« = 0 (S.6)I 2 3

from which it follows by th e above procedure that (3 = f ( ~ 3 ) ,where ~ 3 l i e sbetween

a an d b Cont inuing th is process out to th e nth derivative yields th e result (8.3). Th el as t t er m of this series, namely

is known as th e remainder after n terms. For t he i mp or ta nt case in which f(x) is afunct ion with con tinuous der ivat ives of all orde rs, (8.3) becomes an infinite series asn co , If th e remainder goes to zero in this process, th e series converges to th e value

f(b) an d one may write

(8.7)

EXAl\1PLE S. l

If fex) = sin x, th e remainder does go to zero and th e expansion (8.7) is applicable. If onelets a = 1r 4 an d b = 1r 6 i t follows that f(a) = 1/ y 2 an d feb) = t. (8.7) gives

1 r 1 1r)2 1 1r)3 ]f(b) = V2 1 - 12 2 12 + 6 12 + . . .

Use of onl y th e first four terms of this series gives th e approximation

f(b) ~ 0.4999

S.3 TAYLOR S SERIES FOR ONE VARIABLE

If f(x) an d all i ts der iva tives are continuous in th e interval X l X X2, an d if a an d xa re a ny t\VO point s within thi s interval, it follows from (8.3) that

(S.8)

in which ~ n is some point between a and x. If

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SECTION 8.4

for all x within XI,X2], then

Taylor 's Series for Several Variables 561

~ (x - a)mf(x) == , fm a

m= O n ~ .(8.9)

is a convergent series representation for f(x), valid within th e entire interval. (8.9) isknown as th e Taylor s series expansion of f(x) about t he p oi nt a.

The special case of this resul t in which a = 0 is known as Maclaurin s series, an d can

be written00

\ xm

f(x) = L - jm O)m= O m

(8.10)

Another useful form of Taylor s series results when f( x t ~ x is expanded in a seriesa bo ut t he point x. A straight substitution in (8.9) gives

~ ( ~ x ) mf(x + ~ x = L _ , _ j m x

m= O 1n.

Both x an d x + Lix must be within th e interval XI,X2].

(S.11 )

EXAMPLE 8.2Consider the function f(x + ~ x = (x + Lix)n in which n is an integer. Then f(x) = xn an d

Substitution in (S.ll) gives

n'fm(x) = . xn - m

(n - m)

fm(x) = 0 m > n

n

(x + ~ x ) n= \ n x n - m ( ~ x ) mm

o m (n - m)

n( n - 1)= z + n x n - l ~ x+ x n - 2 ~ x 2+ + n x ~ x n - l+ ~ x n

2

which can be recognized as th e binomial expansion.

S.4 TAYLOR S SERIES FOR SEVERAL VARIABLES

(S.12)

Th e resul ts of th e previous section ma y be extended to functions of more than onevar iabl e with l it tl e difficulty. Le t j(x,y) be an y function which, t ogethe r with all itspartial derivatives, is continuous in th e interval Xl ~ X ~ X2, YI ~ Y ~ Y2 If (a,b) an d(x,y) ar e an y two point s within thi s int erva l, then by Equation (8.9),

j (x ) = ~ (x - a)m amf(a,y)

,y L , a mm= O m. x

(8.13)

But th e functions of y appearing on th e right side of (8.13) also ca n be expanded in a

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562 Taylor's Series

Taylor s series, namely,

l\IATHEl\IATICAL SUPPLEIVIENT: PART I

so that

(8.14)

(S.15)

All th e ser ies in (8.13) , (8.14) , an d (8.15) rnust converge for all points in th e interval

in order for this to be a valid procedure. When they do, (8.15) is known as a 'I 'aylor 'sseries expansion of f(x,Y) about th e point (a,b).

A useful alternative form of (8.15) arises when f( x + LlX, Y + ~ y ) is expanded in aTaylor s series about (x,y). Direct substitution in (S.15) gives

~ L ~ ( ~ x ) m( ~ y ) nam+nj(x,Y)f( x + LlX, Y + /1y) = L - - - - - - -

m =O n =O nd n axmayn(8.16)

N ex t, l et (x,y,z) be an y function which, together with all its partial derivatives, iscontinuous in th e interval Xl X X2, Yl S; Y s 1}2, ZI Z ~ Z2. If a,b,c) an d (x,Y,z)ar e an y t\VO point s within thi s interval, then by (S.15),

_ ~ ~ (x - a)m (y - b)n am+nj(a,b,z)f(x,Y,z) - L L

m = O n = O 1n n axmaynwhereas f rom (8.9) ,

am+n.f(a,b,z) = ~ (z - c)p am+n+pf(a,b,c)

axmayn ~ o p axmaynaz p

Combination of these results gives

_IoI ~ Io

(x - a)m (y - b)n (z - c)p a m+n +7>(a,b,c) (x,Y,z) -

1n n p axmaynaz pm = O n = O p = O

(8.17)

(8.18)

(8.19)

When it is assumed that th e necessary convergence conditions ar e met, (8.19) is known

as t he Taylor s series expansion of (x,Y,z) a bo ut t he point (a,b,c).In an alternative form,

< _ ~ (6.x)m (6.y)n (6.z)p am+n+p (x,y,z)j(x + LlX, Y + ~ Y ,. wI + LlZ) - L L L , , ,

m = O n= O p= O 1n. n. p. ax may71az p (8.20)

The extension of these results to functions of four or more variables follows th e same

procedure an d can be predicted by inspection.

EXAMPLE 8.3In a vacuum triode, th e plate current ib is a function of both th e plate voltage eb and the

grid vol tage e.. In many applications th e triode has a plate current which consists of atime-independent, or d.c. component, an d a t ime-varying component. 'The plate current

c an t he n be expressed in th e formib = Ib + i»

in which Ib is th e quiescent value an d i p is th e superimposed time-varying part. These t\VOcomponent currents flow in response to the voltages eb = Eb + ep an d ec = E e + eg, with

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SECTION 8.4 Taylor's Series for Several Variables 563

(Eb,EJ th e quiescent portions an d ep,e ll ) th e time-varying portions. When Equation (8.16)is applied to this s i tuation, one obtains

If the triode is bia sed to operate in the linear portion of its characteristic, then all higherorder derivatives vanish an d this expansion simplifies to

(8.21)

If one defines t he p la te conductance gp and transconductance gm by the relat ions

the t ime-varying part of (8.21) can be written

(8.22)

Equation (8.22) is the basis for a variety of equivalent circuits for th e t riode which aredistinguished by assumptions concerning th e waveforms of th e signal voltages an d th e

lumped elements placed in th e grid an d plate circuits.

REFERENCES

1. Cajori, F., A History of Miuhenuiiics, 2d ed., pp. 226-229, Th e Macmillan Company, NewYork, 1919.

2. Love, C. E., an d E. D. Rainville, Differential and Integral Calculus, 6t h ed., pp. 439-447,

The Macmillan Company, New York, 1962.3. Smith, D. E., History of Mathematics, vol. 1, pp. 449-454, Ginn an d Company, New York,

1923.