1550251515-Classical_Complex_Analysis__Gonzalez_

596 Chapter^8

where

An= _1 J f(t)dt

27fi (t -a)n+l

(8.18-7)

C1

Similarly, for t E C 2 we have

1 1

t-z (t-a)-(z-a)

1 1

00

(t-ar-^1

= - z - a 1-(t - a)/(z -a) = - ~ (z - a)n

oo ( )-n

"'"" z-a

= - L.; (t -a)-n+l

n=l

where the series representation is now valid for t E C2, since

I

t-al= r' <l

z-a p

From (8.18-8) it follows that

f(t) = _ ~ f(t) (z -a)-n

t - z L.; (t - a)-n+l

n=l

(8.18-8)

(8.18-9)

But f(t) is analytic fort E C 2 , hence continuous. So, there is a constant

M2 > 0 such that lf(t)I < M2 for all t E C2, and we have

I

(z-a)-n I M^2 (r')n

f(t) (t - a)-n+l < 7 p

which shows, by the Weierstrass M-test, that for each z the series in (8.18-

9) converges uniformly with respect to t for t E C2. Thus, again by

Corollary 8.1, we have

_1 J f(t)dt = _ ~ (z -a)-n J f(t)dt

27fi t -z L.; 27fi (t - a)-n+l

C2 n=l C2

00

= -I: A-n(z -a)-n (8.18-10)

n=l

where

A __ 1 J f(t)dt

-n - 27fi (t - a)-n+l

C2

(8.18-11)