is a bijection on the extended complex plane $$\mathbb{C} \cup \{\infty\} given by
$$ f(z) = \begin{cases} \frac{a}{c} &\text{if } z = \infty \\ \infty &\text{if } z = -{d \over c}\\ \frac{az+b}{cz+d} &\text{otherwise} \end{cases}
where $$a,b,c,d \in \mathbb{C} and $$ad - bc \ne 0
It can be shown that the inverse, and composition of two M\"obius transformations are similarly defined, and so the M\"obius transformations form a group under composition.