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Homework Help: Complex analysis question

  1. May 10, 2010 #1
    1. The problem statement, all variables and given/known data
    Q. (a) State Liouville's Theorem

    (b) Suppose that f is analytic in C and satisfies f(z + m + in) = f(z) for all integers m,n . Prove f is constant.

    2. Relevant equations

    3. The attempt at a solution
    (a) Liouville's Theorem - If f is bounded and analytic in C, then f is constant.

    (b) I'm guessing I need to use Liouville's Theorem here i.e. need to show the f is bounded. But I'm so confused! The question states that f(z + m + in) = f(z), so f gives the same value regardless of the z. Doesn't this mean that f is constant?!

    Could someone please point me in the right direction? Thanks for any help :)
  2. jcsd
  3. May 10, 2010 #2


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    Staff Emeritus
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    It doesn't quite say f(z) is constant. It tells you, for example, that

    f(0) = f(1) = f(2) = f(3) = ...


    f(1/2) = f(3/2) = f(5/2) = ...

    but it doesn't directly tell you anything about how f(0) and f(1/2) are related.
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