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Looking for an analytic mapping theorem

  1. Feb 3, 2012 #1
    Say we have a complex function f, analytic on some punctured open disk D\{a} where it has a pole at a. Is there some theorem which says something like: f must map D\{a} to a horizontal strip in ℂ of at least width 2π, or something like that?
     
  2. jcsd
  3. Feb 3, 2012 #2

    morphism

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    There can be no such theorem. Think about what f(z)=1/z does in a punctured disc D\{0}: the image f(D\{0}) won't contain any strips.

    What is true is that if f has a pole at a then you can find a sufficiently small open disc centered at a such that f(D\{a}) contains the complement of an arbitrarily large open disc. This essentially follows from studying the example f(z)=1/z above together with an application of the open mapping theorem.
     
  4. Feb 3, 2012 #3
    ok, thanks morphism.
     
  5. Feb 4, 2012 #4

    morphism

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    I think I might've misinterpreted your question. In my first reading I assumed you were looking for strips centered about the x-axis, but now I see that you didn't make any such requirement. Anyway, as you can see from my previous post, f(D\{a}) will always contain arbitrarily wide strips, if you allow them to be 'high up' the y-axis.
     
  6. Feb 5, 2012 #5
    Ok that would work for me, although I cannot see how this follows from the open mapping theorem, could you explain further how such a result would be derived? Thanks.
     
  7. Feb 11, 2012 #6

    morphism

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    Sorry for the late reply, been busy..

    Anyway: consider g(z)=1/f(z) in D. Since f is holomorphic on D\{a} and has a pole at a, then g is holomorphic on all of D (i.e. the singularity at a is removable). Now apply the open mapping theorem to g.
     
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