1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 453

instance, as seen in the example following Corollary 7.4, we have

f(z~za)n=O·

c

for n = 2, 3, ... , C: z-a = reit, r > 0, 0 St S 21T. Here f(z) = (z-a)-n

is analytic in C - {a}, but fails to be analytic at a E Int C*

7.11 Generalizations of the Cauchy-Goursat Theorem

Theorem 7 .13 If f is analytic in the region R bounded by a simple closed

contour C, and if f is continuous on C, then

J f(z)dz = 0
c
This is called the strong form of the Cauchy-Goursat theorem, first
proved by S. Pollard [26]. For a proof the reader is referred to M. H. A.
Newman [24], 187-188, or to Pollard's paper. However, we shall prove the
following special case.

Theorem 7.14 If f satisfies the same conditions as in Theorem 7.13, and

if the region R bounded by the simple closed contour C contains a point b
such that every ray with origin at b meets C* at just one point, then

J f(z)dz = 0

c

Proof Suppose that C is described once in the positive direction and let

z = b+ p(8)^0 ::;^8 ::; 21T

be the equation of C in polar coordinates with pole at b, and p(21T) = p(O).

Then C and Cr: z = b+rp(8), 0 < r < 1, are homotetic with respect to b,

and since c; c R, by the Cauchy-Goursat theorem we have

J f(z) dz= 0 (7.11-1)

Cr

Consider a partition P = {8 0 , 81 , ... , 8n} of the interval [O, 21T] and let

Zk be the point of C corresponding to 8k ( k = 0, ... , n). Then the sum

n n

~ f(zk)(zk - Zk-1) = ~ J[b + p(8k)][p(8k) - p(8k-1)]

k=l k=l