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Conformal map

  1. Nov 29, 2011 #1
    1. The problem statement, all variables and given/known data
    Show that every conformal self-map of the complex plane C has the form f(z) = az + b, where a ≠ 0. (Hint: The isolated singularity of f(z) at ∞ must be a simple pole.)


    2. Relevant equations



    3. The attempt at a solution
    I know about conformal self-maps of the open unit disk, but this is about the complex plane. I don't know why the isolated singularity of f at ∞ must be a simple pole. I really don't know how I should start.
     
  2. jcsd
  3. Nov 29, 2011 #2

    micromass

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    All we really need to prove is that [itex]\infty[/itex] is a pole. To prove this, we must show that [itex]\infty[/itex] is neither a removable and neither an essential singularity. Do you know equivalent characterizations of being a removable and an essential singularity to make it easier to see if [itex]\infty[/itex] is one??
     
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