So, basically I want to prove the implicit function theorem for holomorphic functions defined in [LaTeX ERROR: Compile failed] : where f has holomorphic derivatives with respect to z and w, if and the gradient is not zero in (actually we ask that the derivative with respect to the second variable not be zero) then there exist an open set such that and there is a fuction such that and for all we have , moreover is holomorphic in V.

I have everything except that is holomorphic. I know it is complex diferentiable at and has a continous real derivative in all V but I don't know how to conclude that it is holomorphic in all , I need to check that C-R holds but I don't know how to do that.

Thanks in advance