1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 543

Fn(z) = An[f(z) - ao]n are analytic in the same disk lzl < r, it follows

from Theorem 8.9 that F(z) is analytic in lzl < r, and that the coefficients

of any given power zn can be obtained by adding the coefficients of zn in
the Maclaurin series of the Fk(z) (k = 0, 1, 2, ... ). That the Maclaurin
series of each

is obtainable by raising the series E ajzi to the kth power, multiplying by
Ak and combining like terms follows from Theorem 8.7.

Theorem 8.10 i~ what is called an existence theorem. It show that the

power series E bnzn actually exists with a positive or infinite radius of
convergence, but except in very simple cases, the computation of the co-
efficients bn cannot be carried out in practice beyond a few beginning
terms.

Example To expa~d F(z) = esinz in powers of z. Letting w = f(z) =
sinz = z - z^3 /3! + z^5 /5! - ·^00 (lzl < oo) and g(w) = ew = 1 + w/1! +
w^2 /2! + · .. (lwl < oo), we have

F(z)=g(f(z))=esinz=l+ ~! (z- ~: + ~: -.. )

1 ( 3 )2 1 ( 3 )3

+ 2! z - ~! +... + 3! z - ~! +... + ...

= 1 + z + 2!^1 z^2 - (1 3! - 4! 1). z^4 + (2 5! - (3!)2 3) z^5 + ...

1 2 1 4 1 5

=l+z+-z - -z - -z +· ..

2 8 15

valid for lzl < oo.

Theorem 8.11 Let w = f(z) = aiz + a2z^2 + · · · = E:=l anzn, lzl < Ri,

and suppose that a 1 #-0. Then the function f has a single-valued inverse

that is analytic in a neighborhood of the origin and has_ a power series
representation of the form

00

z = f-^1 (w) = L bnwn

n=l

The process of obtaining (8.4-10) from w

reversion (or inversion) of the last series.

for lwl < r (8.4-10)

= """oo .lJn=l anZ n lS called the