Results 1 to 4 of 4

Math Help - Radius of Convergence at a point?

  1. #1
    Super Member
    Joined
    Mar 2006
    Posts
    705
    Thanks
    2

    Radius of Convergence at a point?

    Say I have a power series f(z) on the complex plane. Now I know what it means to find the radius of convergence, I know how to use the test, I know how to take the limit superior and use the formula. But what does it mean when one ask what is the radius of convergence of a power series at a particular point z, say z = 1+i?

    Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    The taylor expansion of a complex f(*) 'somewhere around' a point z_{0} is...

    f(z)= \sum_{n=0}^{\infty}\frac{f^{n}(z_{0})}{n!}\cdot (z-z_{0})^{n} (1)

    In general is z_{0}= |z_{0}|\cdot e^{i\theta}. If z_{0} is not on real axis you can use the substitution w=z\cdot e^{-i\theta} so that the series expansion (1) has only real terms. In particular the radious of cenvergence of the series for f(z) and f(w) is the same...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Mar 2006
    Posts
    705
    Thanks
    2
    So if a power series converges everywhere at a point say 0, then does that also means it converges everywhere at other points as well?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    If a power series converges 'everyhere around' a point z_{0}, that means by definition that it coverges 'everywhere around' any other point which is not z_{0}... obvious ...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 7
    Last Post: March 27th 2011, 07:42 PM
  2. Replies: 1
    Last Post: May 13th 2010, 01:20 PM
  3. Replies: 2
    Last Post: May 1st 2010, 09:22 PM
  4. Replies: 1
    Last Post: November 13th 2009, 06:42 AM
  5. series convergence and radius of convergence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 15th 2008, 08:07 AM

Search Tags


/mathhelpforum @mathhelpforum