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 cot( z ) z 4 dz , where C is a circle of large radius R centered at z = 0, and prove that (4) = 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 cot( z ) z 4 are z = n,n Z . Z C R cot( z ) z 4 dz = + X n =- 2 i Res( z = n ) z = n,n 6 = 0 is a pole of order 1. Hence Res( z = n ) = lim z n ( z- n ) cot( z ) z 4 = lim z n ( z- n ) cos( z ) z 4 sin( z ) = cos( n ) n 4 lim z n ( z- n ) sin( z ) = = cos( n ) n 4 lim z n 1 cos( z ) = 1 n 4 We obtain Z C R cot( z ) z 4 dz = 2 i X n =1 1 n 4 + Res( z = 0) ! = 0 And (4) =- 1 2 Res( z = 0) z = 0 is a pole of order 5....
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This note was uploaded on 09/28/2011 for the course PHYSICS 801 taught by Professor Ivanov during the Fall '10 term at Kansas State University.

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

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