W=z^n- transformation in complex space

[omri3012]

Hallo,

When i regard complex function,

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

z go to infinity?

Thanks
Omri

 

[Office_Shredder]

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