Unformatted text preview: has an analytic antiderivative on C { } . 4. Find all real valued harmonic functions on the plane that are constant on all vertical lines. 5. It is a fact that, if n ∈ Z , then 1 sin 2 z1 ( zπn ) 2 has a removable singularity at z = πn . a) Demonstrate this fact in case n = 0. b) Prove that ∞ X n =∞ 1 ( zπn ) 2 converges uniformly on every bounded set after dropping ﬁnitely many terms. c) Finally, use Liouville’s Theorem to prove that 1 sin 2 z = ∞ X n =∞ 1 ( zπn ) 2 ....
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 Spring '09
 Math, Holomorphic function, A. Eremenko, onetoone conformal map, S. Bell

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