MA530_AUG97 - Z ∞ sin x x dx Hint: Integrate e iz /z...

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MATH 530 Qualifying Exam August 1997 Notation: D 1 (0) denotes the unit disk, { z C : | z | < 1 } . 1. Give a careful statement and proof of exactly one of the following theorems: The Fundamental Theorem of Algebra The Partial Fractions Decomposition Theorem 2. Find an analytic function that maps Ω = D 1 (0) - [0 , 1] one-to-one and onto the vertical strip S = { z C :0 < Re z< 1 } . Is the mapping you found unique? Explain. 3. Prove the following mini-version of the Mittag-Leffler Theorem: There exists an analytic function f ( z )on C -{ n : n =1 , 2 , 3 ,... } such that f ( z ) has a simple pole at z = n with principle part 1 / ( z - n )fo reach n N .(Y o uc a nu s ea n y theorems from Ahlfors except the Mittag-Leffler Theorem.) 4.
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Unformatted text preview: Z ∞ sin x x dx Hint: Integrate e iz /z around the contour below. (Prove any limits you use). C R C ε ε R . 6. Suppose that f is an analytic function on the complex plane minus the two points ± 1. Let γ 1 denote the curve given by z 1 ( t ) = 1 + e it where 0 ≤ t ≤ 2 π and let γ 2 denote the curve given by z 2 ( t ) =-1 + e it where 0 ≤ t ≤ 2 π . Suppose that Z γ j f ( z ) dz = 0 for j = 1 , 2. First, explain why Z γ f ( z ) dz = 0 for any closed curve in C-{± 1 } . Next, prove from first principles that f has an analytic antiderivative on C- {± 1 } , i.e., show that there is an analytic function g on C-{± 1 } such that g = f ....
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