Analytic implicit function theorem I'm looking for a proof of the analytic implicit function theorem (IFT). The only related proof I could find was the holomorphic inverse function theorem (by Henri Cartan). On Wikipedia, the analytic IFT is mentioned casually in the general article "Implicit function theorem", saying that "Similarly, if f is analytic inside U×V, then the same holds true for the explicit function g inside U. This generalization is called the analytic implicit function theorem." Mmmh, that's fast...
A sketch of the proof may be the following :


*

*use analytic continuation to transform f into a holomorphic function

*use the holomorphic inverse function theorem (Cartan) to prove a holomorphic IFT

*restriction : g is holomorphic on $\mathbb{C}$, therefore analytic on $\mathbb{R}$.


But it seems weird and I don't think it would work (I have no idea whether a so-called holomorphic IFT exists or not). What would be an efficient proof of the theorem ? Thanks a lot by advance.
 A: 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 $U\subset \mathbb C$, 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.
A: 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.
A: The implicit function theorem is deduced from the inverse function theorem in most standard texts, such as Spivak's "Calculus on Manifolds", and Guillemin and Pollack's "Differential Topology". Basically you just add coordinate functions until the hypotheses of the inverse function theorem hold.
A: 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).
