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November 1st, 2015, 11:16 PM  #1 
Newbie Joined: Jul 2015 From: India Posts: 7 Thanks: 1  Prove analytic function is constant, if argument is constant
Show that analytic function with constant argument is constant.. Need help with this one, unable to prove.. 
November 2nd, 2015, 05:34 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
This is true by the definition of a function.

November 2nd, 2015, 08:38 AM  #3 
Newbie Joined: Jul 2015 From: India Posts: 7 Thanks: 1 
Agreed, but still need a formal proof. Can you please help?

November 2nd, 2015, 09:10 AM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  
December 4th, 2015, 11:45 AM  #5 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,637 Thanks: 119  
December 4th, 2015, 03:24 PM  #6 
Senior Member Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT 
In this context, I think that argument refers to the value of when expressing a complex number in polar form: .

December 4th, 2015, 08:05 PM  #7  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,637 Thanks: 119  Quote:
From Cauchy Riemann eqs with theta constant, $\displaystyle \partial u/\partial r=\frac{1}{r}\partial v/\partial \theta=0\\ \partial v/\partial r=\frac{1}{r}\partial u/\partial \theta=0 $ So u and v are constant. So f(z)=u+iv=constant.  

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analytic, argument, constant, function, prove 
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