1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 595

Fig. 8.9

For t E Ci we have

1 1

=------

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

1 1 ~ (z - a)n (8.18-4)

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

where the series representation of the right is valid since for t E Ci we have

I

z-a I p

t-a = R' < l

From (8.18-4) it follows that

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

t - z L.J n=O (t - a)n+i

(8.18-5)

Since f(t) is analytic for t E Ci, it is continuous on Ci, so there is a

constant Mi > 0 such that lf(t)I <Mi for all t E Ci. Hence

jict) (;~ -:i)~:i I < ~: ( ~, r

which shows, on applying the Weierstrass M-test, that for each z the series
in (8.18-5) converges uniformly with respect to t for t E Ci. Thus, by
Corollary 8.1, we have

_1 jf(t)dt=~(z-a)nf f(t)dt

27l'i t-z L.J 27l'i (t-a)n+i

01 n=O 01

00

(8.18-6)