Results 1 to 2 of 2

Math Help - Maclaurin and Laurent series in the complex plane.

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    1

    Maclaurin and Laurent series in the complex plane.

    Let f(z) = (z+2)/(z-2)

    Find the Maclaurin series for f on the domain abs(z) < 2. Find the Laurant series for f centered at z0 = 0 on the domain 2 < abs(z) < Inf. And find the Laurant Series for f centered at z0 = 2 on the domain 0 < abs(z-2) < Inf.
    Last edited by mr fantastic; April 19th 2010 at 12:16 AM. Reason: Re-titled post
    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
    Is...

    f(z)= \frac{z+2}{z-2} = 1 + \frac{4}{z-2} = 1 - \frac{2}{1-\frac{z}{2}} =

    = 1 - 2\cdot \sum_{n=0}^{\infty} (\frac{z}{2})^{n} = -1 - z - \frac{z^{2}}{2} - \frac{z^{3}}{4} - \dots (1)

    The (1) converges for |z|<2. Setting in (1) s=\frac{1}{z} we have...

    f(s)= \frac{1 + 2s}{1-2s} = -1 - \frac{1}{s} - \frac{1}{2 s^{2}} - \frac{1}{4 s^{3}} - \dots (2)

    ... and the (2) converges for |s|>2...

    Kind regards

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

Similar Math Help Forum Discussions

  1. Laurent series (Complex)
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: May 15th 2010, 01:03 AM
  2. Complex Analysis: Laurent Series
    Posted in the Calculus Forum
    Replies: 7
    Last Post: December 8th 2008, 04:19 PM
  3. Replies: 4
    Last Post: November 17th 2008, 11:53 PM
  4. Laurent Series - Complex Numbers
    Posted in the Calculus Forum
    Replies: 2
    Last Post: June 17th 2007, 05:15 AM
  5. Laurent Series-Complex
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 20th 2006, 06:54 AM

Search Tags


/mathhelpforum @mathhelpforum