Results 1 to 3 of 3

Math Help - Residue - Solution only for even numbers?

  1. #1
    Junior Member
    Joined
    May 2010
    Posts
    38

    Residue - Solution only for even numbers?

    Hello,
    I'm trying to find residues for f(z) = \frac{(z^2 + 1)^n}{z^{n+1}} .
    A binomial expansion of the numerator gives-
    \frac{1}{z^{n+1}} \sum_{k= 0}^n   \binom  {n}{k} z^{2k} =  \sum_{k= 0}^n   \binom  {n}{k} z^{2k-n-1}
    And so the coefficient of \frac{1}{z} is obtained when k = n/2, Giving a residue: Res = \binom  {n}{n/2}
    My question is: what happens when n is an odd number?
    Last edited by dudyu; May 16th 2011 at 02:04 AM. Reason: typo
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by dudyu View Post
    Hello,
    I'm trying to find residues for f(z) = \frac{(z^2 + 1)^2}{z^{n+1}} .
    A binomial expansion of the numerator gives-
    \frac{1}{z^{n+1}} \sum_{k= 0}^n   \binom  {n}{k} z^{2k} =  \sum_{k= 0}^n   \binom  {n}{k} z^{2k-n-1}
    And so the coefficient of \frac{1}{z} is obtained when k = n/2, Giving a residue: Res = \binom  {n}{n/2}
    My question is: what happens when n is an odd number?
    Is there a typo in the question? It looks as though the numerator should be (z^2 + 1)^n, not (z^2 + 1)^2.

    If so, then in the case where n is odd there will be no 1/z term in the expansion, so the residue will be 0.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    May 2010
    Posts
    38
    Quote Originally Posted by Opalg View Post
    Is there a typo in the question? It looks as though the numerator should be (z^2 + 1)^n, not (z^2 + 1)^2.

    If so, then in the case where n is odd there will be no 1/z term in the expansion, so the residue will be 0.
    Yes, there was a typo. thank you!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. residue
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 23rd 2011, 12:17 PM
  2. Complex numbers: how do I show unique solution?
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: May 8th 2011, 07:27 PM
  3. Replies: 1
    Last Post: March 24th 2010, 12:14 AM
  4. Replies: 2
    Last Post: September 7th 2009, 02:01 PM
  5. Urgent, residue and winding numbers
    Posted in the Calculus Forum
    Replies: 8
    Last Post: November 12th 2008, 12:04 PM

Search Tags


/mathhelpforum @mathhelpforum