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Sol-HW4 - Math Methods Solutions for Assignment 4 Sep 15...

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Unformatted text preview: Math Methods Solutions for Assignment 4 Sep. 15, 2010 Fall 2010 Evaluation of Integrals Due Sep. 22, 2010 1(9). Evaluate the integral: Z C R π cot( πz ) z 4 dz , where C is a circle of large radius R centered at z = 0, and prove that ζ (4) = π 4 90 . Hint 1: As R approaches infinity, the integral approaches zero. Hint 2: Series expansion of the cotangent function near zero is cot( x ) = 1 x- x 3- x 3 45 + ... Singularities of π cot( πz ) z 4 are z = n,n ∈ Z . Z C R π cot( πz ) z 4 dz = + ∞ X n =-∞ 2 πi · Res( z = n ) z = n,n 6 = 0 is a pole of order 1. Hence Res( z = n ) = lim z → n ( z- n ) π cot( πz ) z 4 = lim z → n ( z- n ) π cos( πz ) z 4 sin( πz ) = π cos( πn ) n 4 · lim z → n ( z- n ) sin( πz ) = = π cos( πn ) n 4 · lim z → n 1 π cos( πz ) = 1 n 4 We obtain Z C R π cot( πz ) z 4 dz = 2 πi ∞ X n =1 1 n 4 + Res( z = 0) ! = 0 And ζ (4) =- 1 2 · Res( z = 0) z = 0 is a pole of order 5....
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Sol-HW4 - Math Methods Solutions for Assignment 4 Sep 15...

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