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November 1st, 2015, 11:16 PM   #1
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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..
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November 2nd, 2015, 05:34 AM   #2
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This is true by the definition of a function.
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November 2nd, 2015, 08:38 AM   #3
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Agreed, but still need a formal proof. Can you please help?
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November 2nd, 2015, 09:10 AM   #4
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Quote:
Originally Posted by ag07 View Post
Agreed, but still need a formal proof. Can you please help?
What is your (formal) definition of a function?
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December 4th, 2015, 11:45 AM   #5
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Quote:
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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.
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December 4th, 2015, 03:24 PM   #6
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In this context, I think that argument refers to the value of when expressing a complex number in polar form: .
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December 4th, 2015, 08:05 PM   #7
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In this context, I think that argument refers to the value of when expressing a complex number in polar form: .
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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