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Math Help - complex analysis entire function

  1. #1
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    complex analysis entire function

    Show that an entire function whose real part is positive throughout the complex plane must be a constant.
    Can I get some help please?
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by dori1123 View Post
    Show that an entire function whose real part is positive throughout the complex plane must be a constant.
    Can I get some help please?
    Suppose  f(z) satisfies the properties stated above, let's look at  \frac{1}{f(z)} . This function is also entire since  f(z) \neq 0 .

    Since  f(z) \neq 0, \; 0<a\leq |f(z)| for some  a\neq0 . Thus  |f(z)| is bounded from below by a non zero real number.

    So  \left|\frac{1}{f(z)}\right|\leq\frac1a or in other words,  \left|\frac{1}{f(z)}\right| is bounded from above.

    Thus by Liouville's Theorem  \frac{1}{f(z)} is (a non zero) constant, which implies  f(z) is constant too.
    Last edited by chiph588@; April 14th 2010 at 10:38 PM.
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