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-   -   Prove analytic function is constant, if argument is constant (http://mymathforum.com/complex-analysis/249437-prove-analytic-function-constant-if-argument-constant.html)

 ag07 November 1st, 2015 11:16 PM

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..

 CRGreathouse November 2nd, 2015 05:34 AM

This is true by the definition of a function.

 ag07 November 2nd, 2015 08:38 AM

 CRGreathouse November 2nd, 2015 09:10 AM

Quote:
 Originally Posted by ag07 (Post 438064) Agreed, but still need a formal proof. Can you please help?
What is your (formal) definition of a function?

 zylo December 4th, 2015 11:45 AM

Quote:
 Originally Posted by ag07 (Post 437694) 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.

 Petek December 4th, 2015 03:24 PM

In this context, I think that argument refers to the value of $\theta$ when expressing a complex number in polar form: $z=re^{i\theta}$.

 zylo December 4th, 2015 08:05 PM

Quote:
 Originally Posted by Petek (Post 468746) 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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