Results 1 to 3 of 3

Math Help - F-Theory, Identity Theorem

  1. #1
    Newbie
    Joined
    Mar 2009
    Posts
    12

    F-Theory, Identity Theorem

    Hi,

    I'm a little stuck with this exercise.

    Let f:\mathbb{C}\to \mathbb{C} be a holomorphic function which satisfies f(z)\in\mathbb{R}\quad\forall z\in\mathbb{R}. I want to show that
    f(\overline{z}) = \overline{f(z)}.

    We do have a non-discrete set \mathbb{R} on which this identity holds, so if both functions were holomorphic, the proof was complete using the identity theorem.
    But why are the functions z\mapsto f(\overline{z}) and z\mapsto \overline{f(z)} holomorphic?

    Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    If f(z) il holomorphic \forall z\in\mathbb{C}, it will be true also for z=0, so that f(z) can be written as…

    f(z)=\sum_{n=0}^{\infty} a_{n}\cdot z^{n} (1)

    But f(z) \in\mathbb{R} \forall z\in\mathbb{R}, so that all the  a_{n} in (1) are real. Since  \overline {z^n}= \overline {z}^{n} for (1) is  f(\overline {z})= \overline {f(z)}

    Kind regards

    \chi \sigma



    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2009
    Posts
    12
    Oh, thank you! So I was on the wrong train with wanting to use the theorem.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Basic proof of identity in introductory set theory
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: March 28th 2011, 01:37 PM
  2. [SOLVED] An identity of equivalent products related to the binomial theorem
    Posted in the Differential Geometry Forum
    Replies: 8
    Last Post: September 19th 2010, 11:53 PM
  3. Identity using binomial theorem
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: November 2nd 2008, 09:41 AM
  4. identity theorem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 1st 2008, 12:12 AM
  5. prove trig identity by using commutative ring theory
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: October 21st 2007, 01:31 PM

Search Tags


/mathhelpforum @mathhelpforum