Posted by ubpdqn on August 13, 2012

The Riemannian spherical representation of the complex plane brings wonderful insights into complex functions. The above video illustrates *complex inversion * (as per Needham), i.e.

$latex h: z\mapsto\frac{1}{z}$

This can be decomposed into:

$latex f: z\mapsto \frac{1}{\bar{z}}$ (inversion)

and

$latex g: z\mapsto\bar{z}$ (complex conjugation)

i.e.

$latex h =g\circ f$

On the sphere, the first operation corresponds to reflection in the plane z=0 and the second operation to reflection in the plane y =0. The composition yields a rotation of 180 degrees around the x (real axis).

The cdf of the video is here.

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