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Bounded Function

  1. Dec 4, 2008 #1
    How to show that if f is an entire function,such that f(z) = f(z + 2π ) and f(z) = f(z + 2π i)
    for all z belong to C.
    π is pi.
     
  2. jcsd
  3. Dec 4, 2008 #2

    HallsofIvy

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    How to show "if ...."

    but where is your conclusion? What do you want to prove?
     
  4. Dec 4, 2008 #3
    need to prove f(z) is constant.
    first show f is bounded,then by the Liouville's theorem, f is constant
     
  5. Dec 4, 2008 #4
    let me post the whole question
    Suppose that f is an entire function such that f(z) = f(z + 2π ) and f(z) = f(z + 2π i)
    for all z belong to C. Use Liouville's theorem to show that f is constant.
    Hint: Consider the restriction of f to the square {z = x + iy : 0 <x < 2π ; 0 < y <2π }
     
  6. Dec 5, 2008 #5

    HallsofIvy

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    Looks like a good hint! Although wasn't it [itex]0\le x\le 2\pi[/itex], [itex]0\le y\le 2\pi[/itex]? The "=" part is important because that way the set is both closed and bounded and so any continuous function is bounded on it. Since f is "periodic" with periods [itex]2\pi[/itex] and [itex]2\pi i[/itex], the bounds on that square are the bounds for all z.
     
  7. Dec 5, 2008 #6
    Thank you for your answer..
    I finally know how to use the hint..
    At the begining i really dont know how to start..
     
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