Results 1 to 2 of 2

Math Help - Implicit function theorem for holomorphic functions

  1. #1
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721

    Implicit function theorem for holomorphic functions

    So, basically I want to prove the implicit function theorem for holomorphic functions defined in [LaTeX ERROR: Convert failed] : f:\mathbb{C}^2 \rightarrow \mathbb{C} where f has holomorphic derivatives with respect to z and w, if f(z_0,w_0)=0 and the gradient is not zero in (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 V \subset \mathbb{C} such that z_0 \in V and there is a fuction w: V \rightarrow \mathbb{C} such that w_0=w(z_0) and for all z \in V we have f(z,w(z))=0, moreover w is holomorphic in V.

    I have everything except that w is holomorphic. I know it is complex diferentiable at (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 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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    At my house.
    Posts
    542
    Thanks
    11
    Differentiate to get w'=-f_z/f_w on V.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Implicit Function Theorem
    Posted in the Calculus Forum
    Replies: 0
    Last Post: October 3rd 2011, 10:16 AM
  2. Implicit Function Theorem
    Posted in the Calculus Forum
    Replies: 6
    Last Post: October 15th 2010, 05:24 PM
  3. Replies: 0
    Last Post: November 13th 2009, 05:41 AM
  4. Replies: 0
    Last Post: October 19th 2009, 07:58 PM
  5. Implicit Function Theorem
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 26th 2009, 07:58 PM

Search Tags


/mathhelpforum @mathhelpforum