25 - Complex Maps

Translation Map

Let $a\in \mathbb{C}$ then

Define $f:C_{\infty }\to C_{\infty }$ by

$$f(z)=z+a\text{ for }z\in \mathbb{C}\text{ and }f(\infty )=\infty$$

then $f$ is a continuous bijection

Dilations

Let $b\in \mathbb{C}$ with $b\ne 0$ then

Define $f:C_{\infty }\to C_{\infty }$ by

$$f(z)=bz\text{ for }z\in \mathbb{C}\text{ and }f(\infty )=\infty$$

then $f$ is a continuous bijection

Inversion

Define $f:{\mathbb{C}}_{\infty }\to {\mathbb{C}}_{\infty }$ by

$$f(z)=\frac{1}{z}\text{ for }z\in \mathbb{C}\setminus \{0\}\text{ and }f(0)=0\text{ and }f(\infty )=\infty$$