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Thread: Implicit function theorem for holomorphic functions

  1. #1
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    Implicit function theorem for holomorphic functions

    So, basically I want to prove the implicit function theorem for holomorphic functions defined in $\displaystyle \mathbb{C}^2 $: $\displaystyle f:\mathbb{C}^2 \rightarrow \mathbb{C}$ where f has holomorphic derivatives with respect to z and w, if $\displaystyle f(z_0,w_0)=0$ and the gradient is not zero in $\displaystyle (z_0,w_0) $ (actually we ask that the derivative with respect to the second variable not be zero) then there exist an open set $\displaystyle V \subset \mathbb{C}$ such that $\displaystyle z_0 \in V$ and there is a fuction $\displaystyle w: V \rightarrow \mathbb{C} $ such that $\displaystyle w_0=w(z_0) $ and for all $\displaystyle z \in V$ we have $\displaystyle f(z,w(z))=0$, moreover $\displaystyle w $ is holomorphic in V.

    I have everything except that $\displaystyle w$ is holomorphic. I know it is complex diferentiable at $\displaystyle (z_0,w_0)$ and has a continous real derivative in all V but I don't know how to conclude that it is holomorphic in all $\displaystyle V$, I need to check that C-R holds but I don't know how to do that.

    Thanks in advance
    Last edited by Jose27; May 18th 2009 at 09:38 PM.
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  2. #2
    Super Member Rebesques's Avatar
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    Differentiate to get $\displaystyle w'=-f_z/f_w$ on $\displaystyle V$.
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