1550251515-Classical_Complex_Analysis__Gonzalez_

500 Chapter^7

1. f is defined for ( on a contour C and for z in an open set A: i.e., f

is defined on C x A.

2. lf(z,()I :5 M (a constant) for all (z,() EC x A.

3. For each ( E C, f is analytic in A.

4. For each z E A, f is continuous on C.

Then

F(z) = j f(z, () d( (7.27-1)

c

is analytic in A, and

F'(z) = j :z J(z, () d( (7.27-2)

c

Proof Let ')': u - z = reit,o :5 t :5 27!', with r small enough so that 'Y* U

Int 'Y* CA (Fig. 7.28). Since f, as a function of z, is analytic in A, we have

f(z,() = ~ J f(u,()du

27l'i u - z

(7.27-3)

"(

by the classical Cauchy's formula. Also, we have by (7.21-2),

~f(z,() = ~ J f(u,()du

oz 27l'i ( u - z )2

(7.27-4)

"(

By using (7.27-3) in (7.27-1), we get

F(z) = ~ Jd(J J(u,()du

27l'i u - z

(7.27-5)

c "(

h+h

u\0

A c

Fig. 7.28