Math 621 Homework Assignment 4 Spring 2000

Due: Thursday, March 23

1. Use Green's Theorem to prove a weaker version of Cauchy-Goursat's Theorem for a rectangle:

   Let f be a holomorphic function defined and having a continuous derivative f' in an open set U containing a rectangle R. Then

   

   Recall the statement of Green's Theorem: Let be an oriented piecewise smooth simple path (i.e., each connected component of does not intersect itself) in the plane. Assume that bounds a region D (and has the induced orientation, i.e., each smooth piece of is oriented so that D is on the left as you move along ). Let p(x,y), q(x,y) be two functions which are defined and have continuous partial derivatives in an open set containing D and . Then

   

2. 1. Let D be an open disk in and let f be continuous in D. Suppose that for every closed rectangle R contained in D. Prove that f is holomorphic.
   2. Suppose that f is continuous in all of and holomorphic in . Prove that f is holomorphic everywhere.
3. Let U be an open subset of and a sequence of holomorphic functions which converges, uniformly on compact subsets of U, to a function f. Prove that f is holomorphic in U and that converges, uniformly on compact subsets of U, to f'.
4. Lang page 132 Problem 1: Find the integrals over the unit circle C:

   (a) (b) (c)

5. Ahlfors page 120 Problem 3: Compute under the condition . Hint: Make use of the equations and .
6. Show that the successive derivatives of an analytic function at a point can never satisfy in two ways: (a) Using Cauchy's Estimate. (b) Using Taylor's Theorem.
7. Lang page 132 Problem 3 (modified):
   1. Let f be an entire function, k a positive integer, and let be the maximum of on the circle of radius R centered at the origin. Then f is a polynomial of degree if and only if there exist constants C and such that

      

      for all . (Note: one direction was proven in HW 1 Problem 7).

   2. Ahlfors, page 130 Problem 2: Show that a function which is analytic in the whole plane and has a non-essential singularity at reduces to a polynomial.
8. Lang page 159 Problem 7: Let f be analytic on a closed disc of radius b;SPMgt;0, centered at . Show that

   

   Hint: Use polar coordinates and Cauchy's Formula.

9. Lang page 159 Problem 9 (modified): Let f be analytic and 1:1 on the unit disk D, and let be the Taylor series expansion of f. Show that

   

10. 1. Show that the functions and have essential singularities at .
    2. Let , . Find the set of zeroes of f. Does have any accumulation points? Explain. (See Lang, page 21 for the definition of an accumulation point).

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