Inversion (map)

The map $w=1/z$ , which turns the plane inside out about the unit circle.

The inversion map is $w=1/z$ , sending each nonzero $z$ to its reciprocal. Writing $z=re^{i\theta}$ gives $w=(1/r)e^{-i\theta}$ , so points inside the unit circle go outside it (and vice versa) while the argument reflects across the real axis. Inversion is the one elementary building block of a Möbius transformation that a plain complex linear map $az+b$ cannot produce on its own, and together the two generate every Möbius transformation.

Examples

Inversion sends $z=2$ to $w=1/2$ , and $z=i/2$ to $w=-2i$ .
The unit circle $|z|=1$ maps to itself under inversion, though individual points move: $z=i\mapsto w=-i$ .