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Math Help - Complex analysis

  1. #1
    Junior Member
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    Apr 2010
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    Complex analysis

    Help me please.

    Let f(z)be an entire function :
    f(z)\notin \mathbb{R},  \forall z\in \mathbb{C}.
    Prove that  f is constant. (Don't use Picard Theorem).

    Thanks!
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  2. #2
    Senior Member
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    Mar 2010
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    Beijing, China
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    Re: Complex analysis

    Hint: find a mobius map g that maps the real line to the unit circle C, then the domain of g(f), D={g(f(z))} has no intersection with C. Since D is connected, D is either inside C or outside C, then {1/g(f(z))} is inside C. Anyway this contradicts the maximal module principle.
    Last edited by xxp9; December 17th 2011 at 09:16 AM.
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