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Math Help - Entire bounded function must be constant.

  1. #1
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    Entire bounded function must be constant.



    I can kind of see whats going on, but not well enough to construct a logical progression towards a proof. Any ideas?
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  2. #2
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    This feels like the solution.



    This feels right, but I tend to get things wrong with my ignorance. Verify?
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  3. #3
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    That basic idea (taking the exponential) is exactly what is needed, but some of the details are a bit dubious. You can use inequalities for real numbers, but not for complex numbers. It would be better to say \exp(f(z)) = e^{u+iv} = e^ue^{iv} and therefore |\exp(f(z))| =  e^u (since e^u>0 and |e^{iv}|=1). Then apply Liouville's theorem.
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  4. #4
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    Quote Originally Posted by Opalg View Post
    That basic idea (taking the exponential) is exactly what is needed, but some of the details are a bit dubious. You can use inequalities for real numbers, but not for complex numbers. It would be better to say \exp(f(z)) = e^{u+iv} = e^ue^{iv} and therefore |\exp(f(z))| =  e^u (since e^u>0 and |e^{iv}|=1). Then apply Liouville's theorem.
    Of course. This makes a lot of sense to me. Thank you.
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