Results 1 to 4 of 4

Math Help - uniform convergence in complex plane

  1. #1
    Junior Member
    Joined
    Feb 2008
    Posts
    36
    Thanks
    1

    uniform convergence in complex plane

    To show that \sum_n^{\infty} \frac{z^n}{1-z^n} converges uniformly on compact subsets of D(0,1) (the unit disc centered at the origin), does is suffice to show that the sum converges uniformly on an arbitrary closed subdisc?

    Subsequently, I must find the power series coefficients. How do I do this? Does anyone have general guidelines or a worked example I could analogize?

    Muchas gracias
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by mylestone View Post
    To show that \sum_n^{\infty} \frac{z^n}{1-z^n} converges uniformly on compact subsets of D(0,1) (the unit disc centered at the origin), does is suffice to show that the sum converges uniformly on an arbitrary closed subdisc?
    Yes, because any compact of D(0,1) is included in such a subdisc centered at 0 (distance to the boundary is positive).

    Subsequently, I must find the power series coefficients. How do I do this? Does anyone have general guidelines or a worked example I could analogize?
    The idea is to expand the ratio into a power series ( \frac{1}{1-a}=\sum_{k=0}^\infty a^k) so as to get a series of powers z^m (for some m depending on k,n), and gather the terms with same power m by re-organizing the summation to have the usual \sum_{p=0}^\infty a_p z^p...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2008
    Posts
    36
    Thanks
    1
    So I have this prior result:

    If for each n, \sum_k a_k ^{(n)} z^k converges on a disc having radius r and being centered at 0,

    and \sum_n \sum_k a_k ^{(n)} z^k converges uniformly on compact subsets of the disc,

    THEN, if we rewrite this as \sum_j b_j z^j, the coefficients b_j are given by \sum_n a_j ^{(n)}.



    It seems I should apply this but am unsure how to do so explicitly. For example, how do I write \sum \frac{z^n}{1-z^n} as a double-series to apply the result?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by mylestone View Post
    It seems I should apply this but am unsure how to do so explicitly. For example, how do I write \sum \frac{z^n}{1-z^n} as a double-series to apply the result?
    You should definitely apply this result. As for the expression as a double-series, did you do what my last post said about expanding \frac{1}{1-z^n} in power series? If not, you should...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Uniform Convergence of a Complex Series
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: January 3rd 2011, 01:34 AM
  2. Replies: 1
    Last Post: October 31st 2010, 08:09 PM
  3. Pointwise convergence to uniform convergence
    Posted in the Calculus Forum
    Replies: 13
    Last Post: November 29th 2009, 09:25 AM
  4. Pointwise Convergence vs. Uniform Convergence
    Posted in the Calculus Forum
    Replies: 8
    Last Post: October 31st 2007, 06:47 PM
  5. Uniform Continuous and Uniform Convergence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 28th 2007, 03:51 PM

Search Tags


/mathhelpforum @mathhelpforum