MATH 371 Problem Set 4

(Due Monday, November10)

1. Let G be the circle of radius 1, centered at 0 with positive orientation traversed once. Compute the following integrals:

a)

b)

2. Let G be the circle of radius 5 centered on 0 traversed once in the counterclockwise direction. Compute the following integrals:

a)

b)

3. Suppose that f(z) is analytic on and inside the unit circle . Further suppose that for all z in . Prove that and .

4. Does the series converge or diverge? If it converges, what is its sum?

5. Find the radius of convergence of each of the following:

a) b) c)

6. The Geometric Series gives for all .

a) Use the Geometric Series to find a power series representation for .

b) Let F(z) be the antiderivative of that satisfies F(0) = 0. By integrating the series found in a) Find a power series representation for F(z).

c). What is the radius of convergence of the series found in b)?

d) The series converges by the Alternating Series Test. Find its sum using b).