nLab biholomorphic function

Definition

A biholomorphic function is a holomorphic function which is a bijection on the underlying sets and whose set-theoretic inverse is also holomorphic.

Properties

If $f:U\to V$ is a bijective holomorphic map of open subsets of complexfied Cartesian space ${\mathbb{C}}^n$, then the determinant of the Jacobian $J(f)$ is nowhere $0$, that is $f^{-1}$ is holomorphic.

(See this MO discussion and also this other MO discussion)

References

• Eric Bedford, What is… a Biholomorphic Mapping, Notices of the AMS,volume 59, number 6 (pdf)