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.



Analytic implicit function theorem
One possible reference is ["Holomorphic functions of several variables: an introduction to the fundamental theory"](http://books.google.com/books?id=nDgBsOurnAIC&lpg=PA72&vq=implicit&hl=fr&pg=PP1#v=onepage&q&f=false) by Ludger Kaup and Burchard Kaup (section 8 of chapter 0).


