My Math Forum  

Go Back   My Math Forum > College Math Forum > Complex Analysis

Complex Analysis Complex Analysis Math Forum


Reply
 
LinkBack Thread Tools Display Modes
February 25th, 2018, 06:19 AM   #1
Member
 
Joined: Jan 2016
From: Blackpool

Posts: 97
Thanks: 2

Holomorphic annulus example

let 0<r<R, prove that there is no holomorphic function f on the annulus A(0,r,R) with f'(z)=1/z

For this question i said that f is equal to the prinicipal branch of a logarithm
log|z|+iarg(z) but I don't know where i can go from here. Another way i was thinking of doing it is to consider cauchys theorem for primatives and finding a suitable circle, thanks.
Jaket1 is offline  
 
February 25th, 2018, 06:45 AM   #2
Member
 
Joined: Jan 2016
From: Blackpool

Posts: 97
Thanks: 2

edit: we know that the prinicipal branch is holomorphic unless z is real and non positive. So if we express z in rectangular form z=x+iy when y=0 we have z=x where z is not holomorphic when x is less than 0. Therefore no matter how small you choose the annulus with centre 0, and 0<r<R, there will always be points on the horizontal axis of the argand plane where the principal branch is not holomorphic.
Is this right?
Jaket1 is offline  
February 25th, 2018, 04:14 PM   #3
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 416
Thanks: 230

Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
Originally Posted by Jaket1 View Post
edit: we know that the prinicipal branch is holomorphic unless z is real and non positive. So if we express z in rectangular form z=x+iy when y=0 we have z=x where z is not holomorphic when x is less than 0. Therefore no matter how small you choose the annulus with centre 0, and 0<r<R, there will always be points on the horizontal axis of the argand plane where the principal branch is not holomorphic.
Is this right?
This doesn't really work since the claim "log is not holomorphic" is exactly what this question is asking you to prove. Try computing a line integral for 1/z and applying Cauchy's theorem.
SDK is offline  
Reply

  My Math Forum > College Math Forum > Complex Analysis

Tags
annulus, holomorphic



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Show the Annulus is polygonally path connected. Jaket1 Complex Analysis 1 January 26th, 2018 06:33 AM
How to prove formula area of annulus Happy Calculus 3 December 15th, 2014 03:06 AM
Proof: if f holomorphic then f(z)=?z+c? Seijo Complex Analysis 0 September 11th, 2012 04:11 PM
Help with a holomorphic function JamesKirk Complex Analysis 0 January 29th, 2012 11:52 AM
Interessting holomorphic functions alpar_r Complex Analysis 3 October 27th, 2010 01:40 PM





Copyright © 2018 My Math Forum. All rights reserved.