MATH 371 – Solutions to Problem Set 4

1. a) On the circle, G, . Thus by Cauchy’s Integral Theorem.

b) First compute by Cauchy’s Integral Formula. Then .

2. a) Let and be circles of radius 0.5 centered on 1 and -1 respectively, each traversed in the counterclockwise direction. Then .

. Let and .

b) Let , , and g be circles of radius ¼ centered on 1, 0 , and -1 with positive orientation. Then

.

Let .

Then .

3. Let G be the unit circle oriented positively. Then by the Cauchy Integral Formula

and

4. Since the series is a geometric series and the series converges. It converges to .

5. a) ¥

b) Hint: Apply the ratio test to the whole power series.

c) 1

6. a)

b) .

c) R = 1.

d)