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February 25th, 2018, 06:19 AM  #1 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 104 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 logz+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. 
February 25th, 2018, 06:45 AM  #2 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 104 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? 
February 25th, 2018, 04:14 PM  #3  
Senior Member Joined: Sep 2016 From: USA Posts: 628 Thanks: 398 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
 

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annulus, holomorphic 
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