Results 1 to 4 of 4

Thread: Entire bounded function must be constant.

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    195

    Entire bounded function must be constant.



    I can kind of see whats going on, but not well enough to construct a logical progression towards a proof. Any ideas?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Oct 2009
    Posts
    195

    This feels like the solution.



    This feels right, but I tend to get things wrong with my ignorance. Verify?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    That basic idea (taking the exponential) is exactly what is needed, but some of the details are a bit dubious. You can use inequalities for real numbers, but not for complex numbers. It would be better to say \exp(f(z)) = e^{u+iv} = e^ue^{iv} and therefore |\exp(f(z))| =  e^u (since e^u>0 and |e^{iv}|=1). Then apply Liouville's theorem.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Oct 2009
    Posts
    195
    Quote Originally Posted by Opalg View Post
    That basic idea (taking the exponential) is exactly what is needed, but some of the details are a bit dubious. You can use inequalities for real numbers, but not for complex numbers. It would be better to say \exp(f(z)) = e^{u+iv} = e^ue^{iv} and therefore |\exp(f(z))| =  e^u (since e^u>0 and |e^{iv}|=1). Then apply Liouville's theorem.
    Of course. This makes a lot of sense to me. Thank you.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: Nov 29th 2011, 02:08 PM
  2. Why bounded ode solution tends to a constant
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: Oct 16th 2011, 06:32 PM
  3. Bounded entire function a polynomial?
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Mar 3rd 2010, 06:59 AM
  4. Entire function
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Oct 31st 2009, 01:42 AM
  5. entire function
    Posted in the Calculus Forum
    Replies: 6
    Last Post: Nov 12th 2008, 05:44 AM

Search Tags


/mathhelpforum @mathhelpforum