Problem 1534 (difficulty: 9/10)

Stereographic projection (see figure) gives a bijection between points on the unit sphere and the set \(\displaystyle \CC\cup\{\infty\}\).

(a) Under this identification what transformations of the sphere arise from the following complex functions?

\(\displaystyle z\mapsto -z; \qquad z\mapsto \overline{z}; \qquad z\mapsto iz; \qquad z\mapsto \frac1z; \qquad z\mapsto \frac{-1}{\overline{z}}; \qquad z\mapsto \frac{z-i}{1-iz} \)

(b) What complex functions correspond to rotations of the sphere?

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