Math 621 Homework Assignment 7 Spring 2000

Due: Tuesday, May 2

1. Find the value of the integral for:

   (a) C the circle .

   (b) C the circle .

2. Suppose that , and h(z) has a zero of order 2 at . Prove that

   

3. Compute the integral of the following functions over the circle :

   (a)

   (b)

   (c)

4.   Let P(z) and Q(z) be polynomials and suppose that . Show that

   

   where the sum runs over all singularities of the rational function P/Q. Do this problem in two ways: (1) Directly, without considering the residue at . (2) As a special case of problem 6.

5.   Lang Problem 37 page 190: Let f be analytic on with exception of a finite number of isolated singularities . Define the residue at infinity

   

   for .

   (a) Show that is independent of R.

   (b) Show that the sum of the residues of f in the extended complex plane is equal to zero. (This result is often refered to as The residue Theorem.)

6.   Lang Problem 38 page 190 (Cauchy's Residue Formula on the Riemann Sphere).
7. Basic Exam, September 1998 Problem 5: Compute where C denotes the circle transversed counterclockwise. Hint: Use the residue at infinity (problem 6) to save computations.
8. Basic Exam, January 2000 Problem 7: Let f be a one-to-one holomorphic map from a region onto a region . Suppose that contains the closure of the disk . Prove that for the inverse function is given by

   

9. Let P(z) be a polynomial of degree d and assume that the roots of P are simple. Show that for R sufficiently large:

   

10. Lang page 189 Problem 29: Let U be a connected open set, and let D be an open disk whose closure is contained in U. Let f be analytic on U and not constant. Assume that the absolute value is constant on the boundary of D. Prove that f has at least one zero in D. Hint: Consider with .
11. Basic Exam January 2000 Problem 9: Prove that the equation

    

    has 3 roots in the unit disk .

12. Basic Exam, August 99 Problem 4: Let be a real number larger that 1. Show that the equation has exactly one solution in the half plane . Moreover, the solution is real.
13. Basic Exam, January 99 problem 3:

    (a)

    Determine the number of zeroes of in the disk .

    (b)

    Compute the integral .

14. (Recommended Problem, you need not handed it in) Lang Page 191 Problem 40 (modified): Let a, with and . Let be the circle of radius R.

    a) Show that there is an analytic branch of defined in and such that is analytic with value 1 at .

    b) Evaluate

    

    c) Evaluate

    

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