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Math Help - complex analysis and Liouville's theorem problems

  1. #1
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    complex analysis and Liouville's theorem problems

    I had these two problems on a problem set and my solutions were incorrect and I haven't been able to figure them out.

    Let f(z) be analytic on the complex plane. Prove f is a constant in the following cases:

    |f(z)|\geq7

    and

    Re f(z)\geq 7, hint: consider \exp(f(z))
    both for all z in the complex plane.

    Thanks.
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  2. #2
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    A non-constant entire function achieves all values in \mathbb{C} at least once with at most one exception: If it's a non-constant polynomial, it attains all values n times, and if it's non-polynomial, it reaches all values infinitely often with at most one exception.
    Last edited by shawsend; December 6th 2009 at 01:44 PM. Reason: added one exception rule
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  3. #3
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    Since \vert f(z)\vert \geq 7 we have that g(z)=\frac{1}{f(z)} is an entire bounded function. For the second one notice that \vert e^{f(z)} \vert =\vert e^{Re(f(z))} \vert \geq e^7 now argue as in the first problem.
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  4. #4
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    Thanks shawsend. Do you or does anyone have something more mathematical? I was half-expecting a Cauchy integral to be used or something. I have some other problems like this and I'm sure I'd be asked something like this for the final where I'd be expected to be more rigorous.
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  5. #5
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    Quote Originally Posted by Jose27 View Post
    Since \vert f(z)\vert \geq 7 we have that g(z)=\frac{1}{f(z)} is an entire bounded function. For the second one notice that \vert e^{f(z)} \vert =\vert e^{Re(f(z))} \vert \geq e^7 now argue as in the first problem.
    Yeah, I inverted the function like you did but apparently it's not complex analytic according to the person who graded my problem set so it was deemed incorrect.
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  6. #6
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    Can anyone verify that g(z)=\frac{1}{f(z)} is indeed analytic based on the above?
    Thanks.
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  7. #7
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    I felt Jose did a better job than me showing that under those constraints, the function must be constant. Everything he did is completely analytic and entire. I don't see what the problem was.
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  8. #8
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    Okay great, thanks to the both of you.
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