Biholomorphism

Summary

A biholomorphism is a bijective holomorphic map between open subsets of $\C^n$ whose inverse is also holomorphic. Two domains related by a biholomorphism are biholomorphically equivalent: they are the same for complex analysis. In one variable every simply connected proper domain of $\C$ is biholomorphic to the disc, by the Riemann mapping theorem. In several variables this fails: for $n \ge 2$ the unit ball and the unit polydisc are not biholomorphic.