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.

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