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Math Help - More Complex Help

  1. #1
    Junior Member ginafara's Avatar
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    More Complex Help

    I have been given this problem...

    f(z) = u(x, y) + iv(x, y) is entire and |u(x, y)| < m for all points
    (x, y) in the xy plane. Prove that u(x, y) is constant.

    I know that since f(z) is entire if f(z) were bounded it would satisfy Liouville's theorem and it would be constant.

    I have thought about this problem a lot and I am thinking I need to show this using the fact that iv(x,y) is the harmonic conjugate of u but HOW???

    Any help would be greatly appreciated.
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by ginafara View Post
    I have been given this problem...

    f(z) = u(x, y) + iv(x, y) is entire and |u(x, y)| < m for all points
    (x, y) in the xy plane. Prove that u(x, y) is constant.

    I know that since f(z) is entire if f(z) were bounded it would satisfy Liouville's theorem and it would be constant.

    I have thought about this problem a lot and I am thinking I need to show this using the fact that iv(x,y) is the harmonic conjugate of u but HOW???

    Any help would be greatly appreciated.
    HINT: What can you say about the function g(z) = e^{f(z)} ?
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