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MA530_AUG10

# MA530_AUG10 - has an analytic antiderivative on C 4 Find...

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MATH 530 Qualifying Exam August 2010 (S. Bell, A. Eremenko) Each problem is worth 20 points 1. Compute Z 0 x sin x 1 + x 2 dx . Hint: Use a rectangular contour and let the dimensions go to infinity one at a time. If you claim that a certain term goes to zero, prove that it does. 2. Find a one-to-one conformal map from the strip { z : 0 < Im z < 1 } onto the half-strip { z : 0 < Re z, 0 < Im z < 1 } . (You may express your answer as a composition of more elementary maps.) 3. Prove that the function f ( z ) = 1 /z does not have an analytic antiderivative on C - { 0 } . Find all integers n = 0 , ± 1 , ± 2 , . . . such that the function g ( z ) = z n e 1
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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 z-1 ( z-πn ) 2 has a removable singularity at z = πn . a) Demonstrate this fact in case n = 0. b) Prove that ∞ X n =-∞ 1 ( z-πn ) 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 ( z-πn ) 2 ....
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