Math 5283, HW Set 6, Fall 2006
Due: Wednesday, November 8, 2006
1. Give a sketch of Beardon’s proof of the general form of Cauchy’s theorem.
See pages 142–144 in Ahlfors. Please do not just copy the proof, rather try to
understand it and give a summary of the steps.
2. Same with Artin’s proof of the fact that the integral of a locally exact
differential is zero over any cycle homologous to zero. See Theorem 16 on page
144 in Ahlfors.
3. Page 148, Problem 2 in Ahlfors.
4. Page 148, Problem 3 in Ahlfors.
5. Page 148, Problem 4 in Ahlfors.
6. Page 148, Problem 5 in Ahlfors.
7. Consider Ω = {z∈C:|z|>1}. The complement of Ω is the Riemann
sphere is not connected so Ω is not simply-connected. Give another proof of
this fact using analytic functions.
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