1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/Applications 753

valid at least for lw - w 01 < E. That is, the series (9.16-2) can be reversed

in a sufficiently small neighborhood of z 0 , and the resulting series (9.16-3)

provides in lw - wo I < fa solution to the equation w = f(z).

Proof Formula (9.16-1) can be written in the form

But

1 I (f'(()d(

z = 27ri (!(()-wo)-(w - wo)

c+

!'(() !'(() 1

(9.16-5)

(!(() -wo) - (w - wo) J(() -wo 1 - (w -wo)/[J(() - wo]

• oo J'(() n


• ~ [!(()-wo]n+l (w -wo) (9.16-6)

the series on the right converging uniformly for ( E C, since

I

!'(() I <M

f(() -Wo -

for all ( E C and some constant M > 0. Also,

I

w -wo I< lw - wol < 1

J(()-wo f

Hence, by using (9.16-6) in (9.16-5) we get, upon term-by-term
integration,

00

=zo+ Lbn(w-wot

n=l

where (9.16-4) has been taken into account, and by observing that

zo = ~ j (f'(()d(

27rz J(() -wo

c+

in view of (9.16-1).

Remarks I. The expansion (9.16-3) gives an alternative proof of the

analyticity of z = 1-^1 (w) in a neighborhood of w 0 •

IL Corollary 9.8 justifies the reversion of a power series w = a 1 z +
a 2 z^2 + · · · (a 1 f= 0), as pointed out in Theorem 8.11 (where, for simplicity,