February 23rd, 2009, 01:38 PM  #1 
Newbie Joined: Feb 2009 Posts: 7 Thanks: 0  Analytic
show that if f(z) is analytic and real valued in a domain D then f(z) is constant in D.

February 23rd, 2009, 05:19 PM  #2 
Member Joined: Jan 2009 Posts: 72 Thanks: 0  Re: Analytic
Harmonic conjugates are unique to within constants. The harmonic conjugate of the imagimary part of given function is a constant. Since the imag part is zero, the function is a constant. This uses a powerful result. A direct proof, say from the CR equations, would be better. 
February 24th, 2009, 05:50 AM  #3 
Senior Member Joined: Jan 2009 From: Russia Posts: 113 Thanks: 0  Re: Analytic
Morera's theorem also helps here.

February 24th, 2009, 02:09 PM  #4  
Global Moderator Joined: May 2007 Posts: 6,850 Thanks: 742  Re: Analytic Quote:
 
February 27th, 2009, 09:10 AM  #5  
Member Joined: Jan 2009 Posts: 72 Thanks: 0  Re: Analytic
Good point Quote:
 

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