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Math Help - how??

  1. #1
    Member
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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)  \in D then f is constant on D
    __________________________________________________ ______________
    my ideas:
    f costant implise f prime is zero ?!
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  2. #2
    MHF Contributor

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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 \frac{\partial u}{\partial x}= \frac{\partial u}{\partial y}= \frac{\partial v}{\partial x}= \frac{\partial v}{\partial y}= 0.
    Last edited by Plato; July 20th 2009 at 03:20 AM. Reason: LaTex
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