This preview shows pages 1–2. 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: (1) Show that f ( z ) = 1 z + n =1 2 z z 2 n 2 is an analytic function on C \ N . Show that the singular points are all simple (= order 1) poles and determine their residues. Show that f is 1periodic and odd. Finally show that f ( z ) = cot( z ). Solution: Note first that for fixed z 6 Z the series n =1 2 z z 2 n 2 con verges, as we can find a bound  2 z z 2 n 2  c/n 2 for some constant c > 0. So at least f ( z ) is welldefined in each point. Then on a compact subset K of C \ Z we note there is a constant M such that  z  < M if z K . Thus we can find a bound for all z K and n > M + 1 2 z z 2 n 2 = 2 z ( z n )( z + n ) 2 M n 2 M 2 2 M 1 M 2 / ( M + 1) 2 1 n 2 , Define f j ( z ) = 1 z + j n =1 2 z z 2 n 2 , then clearly f j is a holomorphic function on C \ Z , and f j f as j uniformly on K (as the tail ends can be uniformly (for z K ) bounded by  f ( z ) f j ( z )  X n j +1  2 z z 2 n 2  2 M 1 M 2 / ( M + 1) 2 X n j +1 1 n 2 which converges to 0 as j .) Hence we conclude that f is holomorphic on C \ Z . To prove that n Z is a simple pole for f (with residue 1) we will show that f ( z ) 1 z n has a removable singularity at z = n . Indeed, write f j ( z ) = 1 z + j X m =1 2 z z 2 m 2 = 1 z + j X m =1 1 z m + 1 z + m = j X m = j 1 z m , and we see that f j has a simple pole at z { j, j +1 ,...,j } with residue 1. Thus ( z n ) f j has a removable singularity at z = n for all j , and hence ( z n ) f has a removable singularity at z = n and f has at most a simple pole at z = n . Moreover for j > n we see that lim z n ( z n ) f j ( z ) = 1, while f j f is a holomorphic function at z = n (prove as before), so the residue of f at z = n equals the residue of f j at z = n equals 1. In particular f is not regular at z = n , it has a pole, and by the previous argument this is simple....
View
Full
Document
This document was uploaded on 12/08/2010.
 Spring '09
 Periodicity

Click to edit the document details