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Unformatted text preview: MATH 530 Qualifying Exam August 2009 (S. Bell, A. Eremenko) Each problem is worth 25 points 1. Compute Z ∞ sin( x 2 ) dx by integrating e- z 2 around the contour that follows the real axis from the origin to R , then follows the circle Re it as t ranges from t = 0 to t = π/ 4, and then follows a line back from Re iπ/ 4 to the origin. Let R tend to infinity. You may use the fact that R ∞-∞ e- x 2 dx = √ π without proof. 2. Suppose a 1 , a 2 , . . . , a N are distinct non-zero complex numbers and let Ω denote the domain obtained by removing the union of the closed line segments joining each of the points a k to the origin, k = 1 , 2 , . . . , N . Prove that there is an analytic function f ( z ) on Ω such that f ( z ) N = N Y k =1 ( z- a k ) . 3. For a closed contour γ , let F ( z ) = Z γ 1 w- z dw for z in the open set Ω equal to the complex plane minus the trace of γ . Flesh out the details in the following standard argument....
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- Spring '09
- Math, Complex number, Open set, Holomorphic function