ps6sol

# ps6sol - ACM 95a/100a Problem Set 6 Solutions Prepared by:...

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Unformatted text preview: ACM 95a/100a Problem Set 6 Solutions Prepared by: Zhiyi Li 11/13/2007 Total: 126 points Include grading section: 2 points Problem 1 (3 10 points) Expand f ( z ) = 1 z (1- z ) in Laurent series that converge in the following domains: a) 0 < | z | < 1 , b) | z | > 1 , c) | z + 1 | > 2 . Solution 1 We will use the geometric series sum formula: 1 1- = X n =0 n , | | < 1 (1) a) Here we let z , since | z | < 1. Thus, we can write f ( z ) = 1 z 1 1- z = 1 z X n =0 z n = X n =0 z n- 1 , < | z | < 1 (2) b) Here we let 1 z , since | z | > 1 fl fl 1 z fl fl < 1. We manipulate f ( z ) so that it is written in terms of 1 z : f ( z ) = 1 z 1 1- z = 1 z 1 z 1 z- 1 =- 1 z 2 1 1- 1 z =- 1 z 2 X n =0 1 z n =- X n =0 1 z n +2 , | z | > 1 (3) c) | z + 1 | > 2 fl fl fl 1 z +1 fl fl fl < 1 2 and fl fl fl 2 z +1 fl fl fl < 1. Therefore, 1 f ( z ) = 1 z 1 1- z = 1 z + 1 1- z = 1 ( z + 1)- 1 + 1 2- ( z + 1) = 1 z + 1 1 1- 1 z +1- 1 z + 1 1 1- 2 z +1 = 1 z + 1 X n =0 1 z + 1 n- X n =0 2 z + 1 n ! = X n =1 1- 2 n ( z + 1) n +1 , | z + 1 | > 2 (4) Alternatively, we note that | z + 1 | > 2 is contained in | z | > 1. Thus, our series from part (b). converges in | z + 1 | > 2, f ( z ) =- X n =0 1 z n +2 , | z + 1 | > 2 (5) Problem 2 (10 points) Without determining the series, specify the region of convergence for a Laurent series representing f ( z ) = 1 / ( z 4 + 4) in powers of z- 1 that converges at z = i . Solution 2 We are considering the Laurent series about z = 1 that converges at z = i . The poles of f ( z ) occur where z 4 + 4 = 0, or at z = 4 1 4 e i ( 4 + k 2 ) , = 1 i . Thus, there is a Laurent series for each of the three domains depicted in Figure 1 (each domain boundary intersects two poles). The Laurent series that converges at z = i is associated with the domain 1 < | z- 1 | < 5 . Problem 3 (5 6 points) Classify all the singularities (removable, poles, isolated essential, branch points, non-isolated essential) of the following functions in the extended complex plane a) z z 2 + 1 , b) 1 sin z , c) log(1 + z 2 ) , d) z sin 1 z , e) exp 1 z- i log 1+ z 1- z cos ( z ) ....
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## This note was uploaded on 02/22/2010 for the course ACM 95A taught by Professor Nilesa.pierce during the Fall '06 term at Caltech.

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ps6sol - ACM 95a/100a Problem Set 6 Solutions Prepared by:...

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