HW8 - the class for cot z ), and apply the Residue theorem...

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MATH 6322, Complex Analysis Fall 2011, HW#8 Due date: Monday, Nov. 16, 2011 1. Prove that csc z = 1 z + 2 z + X n =1 ( - 1) n 1 z 2 - n 2 π 2 , z 6 = 0 , ± π, ± 2 π,. .. Hint : Consider f ( z ) = csc z - 1 z . Then it is holomorphic at the origin (i.e. the origin is a removable singularity, and f (0) = 0. Then use the method I did in the class in deriving the expression of cot z . 2. Find the expression of + X n = -∞ 1 n 2 + a 2 where a > 0 is not an integer. Hint : The proposed approach is based on the following observation: For any integer k , Res 1 z 2 + a 2 cos πz sin πz ( k ) = 1 π ( k 2 + a 2 ) which allows to find expression of + n = -∞ 1 n 2 + a 2 . To use this observation in deriving the expression of + n = -∞ 1 n 2 + a 2 , let γ n be the circle of center 0 and radius (2 n + 1) 1 2 (the same curve which I did in
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Unformatted text preview: the class for cot z ), and apply the Residue theorem with it to the function f ( z ) cot πzdz . Using the fact that | f ( z ) | ≤ 1 n 2-a 2 to show that lim n → + ∞ Z γ n f ( z ) cot πzdz = 0 . 3. Page 149, #30, #32 on the textbook 4. Determine which of the following family is normal. (a) F = { z n } ∞ n =1 on the unit-disc D (0 , 1). (b) F = { z n 1 / 2 } ∞ n =1 on the whole complex plane C . (c) F = { f ( z ) ≡ c for some c ∈ C } on whole complex plane C . 6. Let U ⊂ C be a connected open set. Show that the family F = { f | f is holomorphic on U and Re ( f ) > } is normal. 1...
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This note was uploaded on 02/21/2012 for the course MATH 4377 taught by Professor Staff during the Summer '08 term at University of Houston.

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