ans11 - Math 185. Sample Answers to Problem Set #11 Page...

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Unformatted text preview: Math 185. Sample Answers to Problem Set #11 Page 257 1. The function 1 / ( z 2 + 1) is even, is defined everywhere on the real axis, and has only a pole at z = i in the upper half plane. By Theorem 2 on page 243, the residue there is 1 / 2 i . For all z in the semicircle C R (defined on page 253), we have 1 z 2 + 1 1 R 2- 1 , so Z C R dz z 2 + 1 R 1 R 2- 1 = R R 2- 1 for all R > 1. This expression goes to 0 as R , so by the method of Section 71, we have Z dx x 2 + 1 = 1 2 Z - dx x 2 + 1 = 1 2 2 i 1 2 i = 2 . 5. We first compute some residues. Let f ( z ) = z 2 ( z 2 + 9)( z 2 + 4) 2 . By Theorem 2 on page 243 (with p ( z ) = z 2 / ( z 2 + 4) 2 ), we have Res z =3 i f ( z ) = (3 i ) 2 2(3 i )((3 i ) 2 + 4) 2 =- 9 6 i (- 5) 2 = 3 i 50 . By the theorem on page 234, we have Res z =2 i f ( z ) = d dz z 2 ( z 2 + 9)( z + 2 i ) 2 z =2 i = 2 z ( z 2 + 9)( z + 2 i ) 2- z 2 (2 z ( z + 2 i ) 2 + 2( z 2 + 9)( z + 2 i )) ( z 2 + 9) 2 ( z + 2 i ) 4 z =2 i =...
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This note was uploaded on 09/01/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at University of California, Berkeley.

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ans11 - Math 185. Sample Answers to Problem Set #11 Page...

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