This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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, nonisolated 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 ) ....
View
Full
Document
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.
 Fall '06
 NilesA.Pierce

Click to edit the document details