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Posted by ubpdqn on August 13, 2012

Unknown Blogger Mathematica

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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