Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By johnsomeone

Math Help - Analytic function on entire Riemann sphere is constant

  1. #1
    Newbie
    Joined
    Mar 2012
    From
    Sweden
    Posts
    18
    Thanks
    1

    Analytic function on entire Riemann sphere is constant

    According to Wikipedia (under properties) a function which is entire on the whole Riemann sphere must be constant, but I can't quite see why. Is it because the limit as |z| goes to infinity of f(z) exists, and therefore f is bounded?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    146

    Re: Analytic function on entire Riemann sphere is constant

    Basically, yes.
    If f entire on the Riemann sphere, then in particular it's continuous, so |f| is continuous from the compact sphere into the reals, and hence achieves its maximum. Thus |f| is bounded on the Riemann sphere, and so |f| is also bounded on just the complex plane. Then apply Louisville's theorem, and extend continuously to the point at infinity, to show that f must be a constant function.

    You could also think of it in terms of Laurent series:

    Take the Taylor series for f at z=0. Since f is entire, there are no singularities, hence the radius of convergence is infinity. But f analytic at the point at infinity means that f(1/z) is analytic at z=0 - and using that original power series for f you can see that that means that only the constant term can be non-zero.

    In a Laurent series, the z-powers are to the point at infinity (i.e. "bad" points, singularities) as the (1/z)-powers are to the point z=0 (i.e. "bad" points, singularities). Thus for a Laurent series to have no bad points (meaning it represents an entire function on the whole Riemann sphere), it can't have any non-zero (1/z)-powers (since it's analytic at z=0) and it can't have any non-zero z-powers (since it's analytic at z=the point at infinity). Thus the only non-zero term it can have is the constant term.
    Last edited by johnsomeone; October 17th 2012 at 12:22 PM.
    Thanks from thesmurfmaster
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: November 29th 2011, 02:08 PM
  2. Proving that an analytic function on a region is constant.
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 18th 2011, 05:42 PM
  3. Entire bounded function must be constant.
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 3rd 2010, 09:17 AM
  4. Is this correct? - showing analytic function is constant.
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: February 9th 2010, 04:46 PM
  5. Replies: 2
    Last Post: January 20th 2009, 08:14 PM

Search Tags


/mathhelpforum @mathhelpforum