My Math Forum Prove analytic function is constant, if argument is constant

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 November 1st, 2015, 10:16 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.. Thanks from zylo
 November 2nd, 2015, 04:34 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, 07:38 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, 08:10 AM   #4
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Quote:
What is your (formal) definition of a function?

December 4th, 2015, 10:45 AM   #5
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Quote:
 Originally Posted by ag07 Show that analytic function with constant argument is constant..
The question makes no sense. You are asking what the value of f(z) is when z=c. It is f(c), which has nothing to do with f(z) being constant.

 December 4th, 2015, 02:24 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 $\theta$ when expressing a complex number in polar form: $z=re^{i\theta}$. Thanks from zylo
December 4th, 2015, 07:05 PM   #7
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Quote:
 Originally Posted by Petek In this context, I think that argument refers to the value of $\theta$ when expressing a complex number in polar form: $z=re^{i\theta}$.
Excellent point. Thanks

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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