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W=z^n- transformation in complex space

  1. Jul 12, 2009 #1

    When i regard complex function,

    Why does the transformation w=zn don't "conserve" angels when

    z go to infinity?

  2. jcsd
  3. Jul 12, 2009 #2


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    On the complex plane, at every point except for 0 the function w(z) = zn conserves angles because it's holomorphic with derivative non-zero. So it doesn't matter 'how close' to infinity you are, it's still angle preserving. But on the Riemann sphere, at the point infinity, you have to look at it in a different local coordinate in order to take the derivative. Canonically the coordinate is 1/z, so the point infinity is locally considered 0. In this case AT infinity the function won't preserve angles
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