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Math Help - Complex Analysis - simply connected domain

  1. #1
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    Complex Analysis - simply connected domain

    Please help me to prove.

    Prove that if domain \Omega conformaly maps to unit disc D=\left \{ \left | z \right |<1 \right \} then \Omega it's simply connected domain.

    Thanks!
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  2. #2
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    Re: Complex Analysis - simply connected domain

    Quote Originally Posted by sinichko View Post
    Please help me to prove.

    Prove that if domain \Omega conformaly maps to unit disc D=\left \{ \left | z \right |<1 \right \} then \Omega it's simply connected domain.
    You need to assume that the conformal map f:\Omega\to D is bijective (and therefore invertible), otherwise the result is not true.

    Suppose that S is a loop in \Omega. The image f(S) of S is a loop in the simply-connected space D. So there is a continuous transformation in D that contracts this loop to a point. The image in of that deformation under the map f^{-1} will be a continuous transformation in \Omega which contracts S to a point S to a point.
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  3. #3
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    Re: Complex Analysis - simply connected domain

    [QUOTE=Opalg;696240]You need to assume that the conformal map f:\Omega\to D is bijective (and therefore invertible), otherwise the result is not true.


    Thanks a lot!
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