But you don't know that f is constant, that's what you want to prove. So you might want to use the converse: if f'= 0for all (x,y)then f is a constant. And you can do that using the Cauchy-Riemann equations to show that .

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- Jul 19th 2009, 09:44 PM #1

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## how??

suppose that :

f(z)= u(x,y) + i v(x,y) is an analytic function on a domain D . prove that if thee are real constants a,b,c (not all zeros ) with :

a u(x,y) + b v(x,y) =c for all (x,y) D then f is constant on D

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my ideas:

f costant implise f prime is zero ?!

- Jul 20th 2009, 03:00 AM #2

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But you don't know that f is constant, that's what you want to prove. So you might want to use the converse: if f'= 0

**for all (x,y)**then f is a constant. And you can do that using the Cauchy-Riemann equations to show that .