W=z^n- transformation in complex space

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SUMMARY

The transformation w = zn in complex analysis conserves angles at every point except for 0 due to its holomorphic nature and non-zero derivative. However, when approaching infinity on the Riemann sphere, the local coordinate changes to 1/z, which alters the behavior of the function. Consequently, at infinity, the transformation does not preserve angles. This distinction is crucial for understanding the behavior of complex functions in different contexts.

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omri3012
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Hallo,

When i regard complex function,

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

z go to infinity?

Thanks
Omri
 
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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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