1550251515-Classical_Complex_Analysis__Gonzalez_

544 Chapter^8

Proof Since a 1 = f'(O) f:. 0, the existence and analyticity of a single-valued
inverse in a neighborhood of the origin will follow from Corollary 9.8 (to
be discussed later).
The method of undetermined coefficients may be used to find succes-
sively the coefficients 'bi, b 2 , •••• By raising (8.4-10) to the second, third,
powers, we get

z^2 = biw^2 + 2b1b2w^3 + (b~ + 2b1ba)w^4 + (2b1b4 + 2b2ba)w^5 + · · ·

z^3 = biw^3 + 3bib2w^4 + (3b1b~ + 3biba)w

5

+ · · ·

z4 = bfw4 +

Hence substitution of (8.4-10) into w = I:::'=l anzn gives

w = aib1w + (a1b2 + a2bi)w^2 + (a1ba + 2a2b1b2 + aabi)w^3

+ (a1b4 + a2(b~ + 2b1ba) + 3aabib2 + a4bf)w

4

+ (a1bs + 2a2(b1b4 + b2ba) + 3aab1(b~ + biba) + 4a4bib2

+asbDw

5

+ ···

and equating the coefficients of like powers of w, we obtain successively

1 a 2

bi=-, bz=-3,

ai a 1

.b4 = ----"'--~-----"---5a~ + 5a1a2aa - aia4

aI

(8.14-11)

14a~ - 2la1a~a3 + 6aia2a4 + 3aia~ - aias

bs= -~---"----=------"--""---0...-

ai

The radius of convergence of the resulting series is, of course, the distance

from the origin to the nearest nonremovable singularity of f-^1 ( w ).

Remarlk If the given series is of the form w = f(z) = I:::'=o anzn, lzl <

Ri with Wo = f(O) = ao f:. 0, ai f:. 0, we may write w - Wo = I:::'=l anzn,

and the reversion of the last series leads to a Taylor expansion of the form

z = I:::'=l bn(w - wo)n valid for lw - wol < r.

Example Let w = ez - 1 = z/1! + z^2 /2! + z^3 /3! + · · ·, lzl < oo. Here

an = 1/n! (n = 1, 2, ... ). Using formulas (8.4-11) to obtain the first five

coefficients of the reversed series, we have

1

b2 = --,

2

1

ba = 3'

1

b4 = --,

4

1

bs = -,

5