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 $f(\infty) = \infty$.