Results 1 to 4 of 4

Math Help - Showing a function is constant

  1. #1
    Super Member
    Joined
    Feb 2008
    Posts
    535

    Showing a function is constant

    Let f be analytic on a domain D and suppose that f'(x) = 0 for all z e D. Show that f is constant on D.

    Can someone show how to show this please. I have an exam tomorrow and I need to know how to do this...

    Thanks in advance for any help
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    May 2008
    Posts
    37
    It's a little trickier than the real-variable problem, where you could just use the mean-value theorem.

    Here the trick is to fix z_0 in D (which you have to assume is connected) and let w_0 = f(z_0). Let A = {z in G | f(z) = w_0}. You can show A = D by showing that A is both open and closed in D. The details are messy... I recall there is a pretty good proof in Conway though.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Feb 2008
    Posts
    535
    What is Conway? Can anyone show how this proof is done...

    Thanks
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    You may find this thread informative.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: November 29th 2011, 03:08 PM
  2. Replies: 0
    Last Post: October 19th 2011, 05:49 AM
  3. Showing that a function is odd
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: April 1st 2011, 12:19 AM
  4. Showing a function is one-to-one or onto.
    Posted in the Discrete Math Forum
    Replies: 9
    Last Post: April 13th 2010, 07:58 AM
  5. Is this correct? - showing analytic function is constant.
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: February 9th 2010, 05:46 PM

Search Tags


/mathhelpforum @mathhelpforum