1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 541

Then the coefficients in every column above form a convergent series,

and if we let

we have

00

aok + alk + · · · + amk + · · · = L amk = bk

m=O

00

F(z) = Lbkzk

k=O

at least for JzJ < R.

Proof By Corollary 8.5 the function F(z) is analytic in Jzl < R, and

00

p(k)(z) = L l$:>(z), JzJ <R

m=O

so that, in particular for z = 0,

00

p(o) = 2:: 1$:>(0)

m=O

Now, by the Cauchy-Taylor expansion theorem, the coefficient of zk in the

expansion of F( z) about the origin is

;, p(k>(o) = ;, f 1$:>(0) = f ;, 1$:>(0) = f amk

m=O m=O m=O

=bk

since amk = l~)(O)/k!.

Example Consider the Lambert series

oo m oo

F(z) = """" L.,; _z_ 1 - zm = """"l L.,; m(z)

m=l m=l

This series is uniformly convergent for JzJ _:::; r < 1. In fact,

so that

(8.4-7)