Results 1 to 4 of 4

Math Help - How to show that an entire function is a constant function using Liouville's Theorem?

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    26

    How to show that an entire function is a constant function using Liouville's Theorem?

    we have:
    f is entire such that f(z) = f(z + 2Pi) and f(z) = f(z+2*i*Pi) for all z in the complex plane.
    need to show that f is constant.

    By Liouville's them, we only need to show f is bounded.

    there's a hint which says to restrict f to the square S = {z=x+iy: 0=<x=<2Pi, 0=<y=<2pi}

    ------------------------
    my working so far:

    to show f is bounded maybe look at x=0, y=0, x=2Pi, y=2Pi? i'm not sure what this would yield though
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78

    Re: How to show that an entire function is a constant function using Liouville's Theo

    Quote Originally Posted by cassius View Post
    we have:
    f is entire such that f(z) = f(z + 2Pi) and f(z) = f(z+2*i*Pi) for all z in the complex plane.
    need to show that f is constant.

    By Liouville's them, we only need to show f is bounded.

    there's a hint which says to restrict f to the square S = {z=x+iy: 0=<x=<2Pi, 0=<y=<2pi}

    ------------------------
    my working so far:

    to show f is bounded maybe look at x=0, y=0, x=2Pi, y=2Pi? i'm not sure what this would yield though
    Hint: The unit square is a compact subset of the complex plane. What can we say about analytic functions on compact sets?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2010
    Posts
    26

    Re: How to show that an entire function is a constant function using Liouville's Theo

    Quote Originally Posted by TheEmptySet View Post
    Hint: The unit square is a compact subset of the complex plane. What can we say about analytic functions on compact sets?
    ...they are uniformly bounded? so liouville's follows?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78

    Re: How to show that an entire function is a constant function using Liouville's Theo

    Quote Originally Posted by cassius View Post
    ...they are uniformly bounded? so liouville's follows?
    Yes and since the function is peroidic is must be bounded on all of \mathbb{C}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How to show that this function is entire
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: June 6th 2011, 10:39 AM
  2. show that f(z) is an entire function
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 31st 2011, 09:35 PM
  3. show the function is an entire function
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 24th 2010, 11:46 PM
  4. Entire bounded function must be constant.
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 3rd 2010, 10:17 AM
  5. Show a function f is constant, given...
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: February 27th 2009, 09:00 PM

Search Tags


/mathhelpforum @mathhelpforum