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Math Help - Proof, bijective aplication and conformal mapping

  1. #1
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    Proof, bijective aplication and conformal mapping

    Hi there. I have to prove that if f:A \rightarrow B it's a bijective and analytic function with analyitic inverse, then f is conformal.

    I think I should prove the angle preserving using the analyticity, but I'm not sure how.
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  2. #2
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    Re: Proof, bijective aplication and conformal mapping

    Since it's bijective there is no critical point. Otherwise if p is a cirtical point, f(p) can be expanded as f(z)=az^k + ... thus f is a k-folder covering on a small disk of D\{p}. To show that f preserve angles, note that its differential df ( the induced linear map at the tangent space) is a linear map:
    df(v) = f' * v. Where the right side is the complex product. And since p is not critical f' is no zero, f'=r e^{i \theta} is a linear map composed by a rotation of \theta followed by a dilation of ratio r, while both the rotation and the dilation preserve angles.
    Translate the above idea into math expressions and you're done.
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