Results 1 to 6 of 6

Math Help - residues

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    6

    residues

    Hi,

    I'm working through an example of using residues to evaluate an improper integral, but I don't see how they arrive at the following initial step:

    x^2 / ((x^2 + 1)(x^2 +4)) = 1/3[ (4 / (x^2 + 1) - (1 / (x^2 + 4)]

    Any help would be appreciated

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Mar 2010
    Posts
    280
    \frac{1}{x^2+1} = \frac{1}{(x+i)(x-i)}=\frac{a}{x+i}+\frac{b}{x-i}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by ihr02 View Post
    Hi,

    I'm working through an example of using residues to evaluate an improper integral, but I don't see how they arrive at the following initial step:

    x^2 / ((x^2 + 1)(x^2 +4)) = 1/3[ (4 / (x^2 + 1) - (1 / (x^2 + 4)]

    Any help would be appreciated

    Thanks
    Please post the entire question. Thankyou.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Apr 2010
    Posts
    6
    Sorry about that. The question is using residues to evaluate the integral from 0 to infinity:


    (x^2 / (x^2 + 1)(x^2 + 4))

    Thanks
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by ihr02 View Post
    Sorry about that. The question is using residues to evaluate the integral from 0 to infinity:


    (x^2 / (x^2 + 1)(x^2 + 4))

    Thanks
    This function has simple poles at values of x such that x^2 + 1 = 0 and x^2 + 4 = 0 ....
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by ihr02 View Post
    Hi,

    I'm working through an example of using residues to evaluate an improper integral, but I don't see how they arrive at the following initial step:

    x^2 / ((x^2 + 1)(x^2 +4)) = 1/3[ (4 / (x^2 + 1) - (1 / (x^2 + 4)]
    That initial step just consists of using partial fractions to write \frac{x^2}{(x^2+1)(x^2+4)} = \frac A{x^2+1} + \frac B{x^2+4}, for suitable constants A and B. (If that looks odd, replace x^2 by a single variable, say x^2=t, to make it look more like a standard partial fractions decomposition.)

    Why the worked example ahould have started in this way is a bit of a mystery. As Mr F points out, the way to solve the problem is to use the residue theorem, and it doesn't seem necessary to use partial fractions in order to find the residues.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. About sum of residues
    Posted in the Number Theory Forum
    Replies: 8
    Last Post: May 18th 2009, 11:54 AM
  2. Set of residues
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: April 29th 2009, 08:10 AM
  3. residues
    Posted in the Calculus Forum
    Replies: 6
    Last Post: February 11th 2009, 08:44 AM
  4. residues 2
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 10th 2009, 08:19 AM
  5. Residues
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 6th 2008, 09:29 AM

Search Tags


/mathhelpforum @mathhelpforum