MATH 371 Problem Set 5

Due Monday, December 1 ,but some students might like to turn it in before the Thanksgiving break.

1. Find and classify the zeros and isolated singularities of the following functions:

2. Classify the behavior of the following functions at ¥:

3. Construct a function f(z) that is analytic in the plane except for isolated singularities and satisfies the following conditions:

a) f has a simple zero at 2, a double zero at -2, and a pole of order 3 at 1.

b) f has an essential singularity at 2 and 2i.

c) f has a removable singularity at 1, a pole of order 2 at 2, and a zero of order 2 at -2.

4. Find functions f(z) and g(z) that are analytic in the plane except that each has a pole at z₀ = 1+i and f(z) + g(z) does not have a pole at 1 + i.

5. Suppose that f(z) has a pole of order m at 1. What is the behavior of at 1? Justify your answer.