Results 1 to 3 of 3

Math Help - Louisville's Theorem?

  1. #1
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347

    Louisville's Theorem?

    The function f is holomorphic on \mathbb{C} and it is given that there exists a K > 0 such that

    |f(z) + f(-z)| \le K for all z.


    What can we say about f?

    I was thinking perhaps Louisville's theorem would explain this, (ie bounded entire functions are constant) but I'm not sure how to prove this. Or perhaps f is an odd function. Any help with this would be appreciated
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by slevvio View Post
    The function f is holomorphic on \mathbb{C} and it is given that there exists a K > 0 such that

    |f(z) + f(-z)| \le K for all z.


    What can we say about f?

    I was thinking perhaps Louisville's theorem would explain this, (ie bounded entire functions are constant) but I'm not sure how to prove this. Or perhaps f is an odd function. Any help with this would be appreciated


    For any function we have f(z)=\frac{f(z)+f(-z)}{2}+\frac{f(z)-f(-z)}{2} (sum of even plus odd functions) ...

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    Quote Originally Posted by slevvio View Post
    The function f is holomorphic on \mathbb{C} and it is given that there exists a K > 0 such that

    |f(z) + f(-z)| \le K for all z.


    What can we say about f?

    I was thinking perhaps Louisville's theorem would explain this, (ie bounded entire functions are constant) but I'm not sure how to prove this. Or perhaps f is an odd function. Any help with this would be appreciated
    On the basis of Liousville theorem You can derive that the even part of f(*) is a constant or alternatively that f(*) is the sum of a constant and an odd function...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: January 10th 2011, 09:51 AM
  2. Replies: 3
    Last Post: May 14th 2010, 11:04 PM
  3. Prove Wilson's theorem by Lagrange's theorem
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: April 10th 2010, 02:07 PM
  4. Replies: 2
    Last Post: April 3rd 2010, 05:41 PM
  5. Replies: 0
    Last Post: November 13th 2009, 06:41 AM

Search Tags


/mathhelpforum @mathhelpforum