Möbius transformation

Definition

Let $a, b, c, d$ be complex numbers with $ad - bc \neq 0$. The Möbius transformation with these coefficients is the map $f : \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ of the Riemann sphere to itself given by $$f(z) = \frac{az + b}{cz + d}$$ for each $z \in \mathbb{C}$ with $cz + d \neq 0$. The map also has the following values. If $c \neq 0$, then $f(-d/c) = \infty$ and $f(\infty) = a/c$. If $c = 0$, then $f(\infty) = \infty$.

Remarks

Each Möbius transformation is a bijection of $\widehat{\mathbb{C}}$, and its inverse is the Möbius transformation $w \mapsto \frac{dw - b}{-cw + a}$. The Möbius transformations form a group under composition, the Möbius group. The coefficients $(\lambda a, \lambda b, \lambda c, \lambda d)$ with $\lambda \neq 0$ give the same map. If $ad - bc = 0$, the formula gives a constant or is undefined, so it is not a Möbius transformation.

Explanations

L1 human

A Möbius transformation is a quotient of two linear functions of $z$. Examples are $z \mapsto z + 1$, $z \mapsto 2z$ and $z \mapsto \frac{1}{z}$, and each Möbius transformation is a composition of maps of these simple types. The point $\infty$ is necessary, because the denominator can be zero.