# Analytic implicit function theorem

One possible reference is “Holomorphic functions of several variables: an introduction to the fundamental theory” by Ludger Kaup and Burchard Kaup (section 8 of chapter 0).

In one variable, this is a trivial consequence of the standard local inversion theorem. Indeed, holomorphic functions are \$C^1\$ functions characterized by the fact that their differential is a similitude. And this property is stable by taking the inverse.

to be more precise, if \$g\$ is holomorphic on some open set \$Usubset mathbb C\$, and its differential (as a function \$U to mathbb R^2\$) satisfies that it is invertible everywhere, with differential being a similitude. So the functions is a local diffeomorphism, and the differential of the inverse is the inverse of the differential, so is still a similitude. Therefore, \$g\$ is a local biholomorphism.

The statement of IFT is a direct consequence of the local inversion theorem then.

There’s a proof of both the analytic inverse function thm and the analytic implicit function thm (where the second is rather “formally” deduced from the first) in the following book:

Fritzsche, Grauert, From Holomorphic Functions to Complex Manifolds

Chapter 7 “Holomorphic maps” (in which both theorems are proved) is freely available online.