Holomorphic automorphisms of C^n

Summary

A holomorphic automorphism of $\C^n$ is a biholomorphism of $\C^n$ onto itself. Under composition these maps form the group $\Aut(\C^n)$, an automorphism group. For $n = 1$ this group consists only of the affine maps $z \mapsto az + b$ with $a \ne 0$. For $n \ge 2$ it is very large, and its study is a central topic of complex analysis in several variables.