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Complex Analysis - Final Examination 10:20AM to 12:20 PM, June 15, 2004 (1) (10 %) Evaluate C e z z 4 + 5 z 3 dz with C : | z | = 2. (2) (20 %) Find the Laurent series of f ( z ) = 1 ( z - 1) 2 ( z - 3) (2a) (10 %) in 0 < | z - 1 | < 2 with center at z = 1, (2b) (10 %) in 0 < | z - 3 | < 2 with center at z = 3. (3) (20 %) Evaluate P . V . -∞ sin x x ( x 2 - 2 x + 2) dx . (4) (20 %) Prove that P . V . 0 x a - 1 1 - x dx = π cot with 0 < a < 1. (5) (10 %) The functio w = e z/ 4 maps a region R in the z -plane to a region R in the
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Unformatted text preview: Find R ± if R = { z = x + iy | ≤ y ≤ π } . (6) (20 %) Apply Schwarz-Christoﬀel formula, f ± ( z ) = A ( z-x 1 ) ( α 1-1) ( z-x 2 ) ( α 2-1) , to ±nd the function w = f ( z ) such that D = { z = x + iy | y ≥ } is mapped to D ± = { w = u + iv | u ≥ , | v | ≤ 1 } , under the conditions that f (-1) =-i and f (1) = i . 1...
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