ans10 - Math 185. Sample Answers to Problem Set #10 Page...

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Unformatted text preview: Math 185. Sample Answers to Problem Set #10 Page 233 1. a . From the example on page 233, we have z exp 1 z = z + 1 + 1 (2!) z + 1 (3!) z 2 + ... with principal part 1 (2!) z + 1 (3!) z 2 + ... . This is an essential singularity. b . The isolated singular point is z =- 1. At that point the Laurent series is z 2 1 + z = ((1 + z )- 1) 2 1 + z = (1 + z ) 2- 2(1 + z ) + 1 1 + z with principal part 1 / 1 + z . This is a (simple) pole. c . We have sin z z = X n =0 (- 1) n z 2 n (2 n + 1)! , with principal part 0. This is a removable singularity. d . We have cos z z = X n =0 (- 1) n z 2 n- 1 (2 n )! , with principal part 1 /z . This is a pole. e . The function 1 / (2- z ) 3 has principal part- 1 / ( z- 2) 3 , which means that it is a pole. 4. Let a be a positive real number, and (as the problem asks) write f ( z ) = 8 a 3 z 2 ( z 2 + a 2 ) 3 = ( z ) ( z- ai ) 3 , where ( z ) = 8 a 3 z 2 ( z + ai ) 3 . 1 2 This is so because z 2 + a 2 = ( z- ai )( z + ai ). The function ( z ) is analytic at z = ai , so it has a Taylor series there:...
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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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ans10 - Math 185. Sample Answers to Problem Set #10 Page...

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