Results 1 to 2 of 2

Math Help - Poisson summation

  1. #1
    Newbie
    Joined
    Aug 2011
    Posts
    16

    Poisson summation

    Hi, I think something is wrong with my working because I have a 2 where it should only be a 1.

    Q: Show

    \sum_{n=-\infty}^\infty \frac{1}{a^2n^2+1} = \frac{\pi}{a} coth \frac{\pi}{a}

    Working:
    I have got the fourier transform of \frac{1}{a^2n^2+1} which is f(k)=\frac{\pi}{a}e^{-2|k| \frac{\pi}{a}}.

    So from here I use:

    \sum_{m=-\infty}^\infty f(m) = \sum_{n=-\infty}^\infty F(2 \pi n)

    Thus:

    \sum_{n=0}^\infty F(-2 \pi n) + \sum_{n=1}^\infty F(2 \pi n) = \frac{\pi}{a} [2 \sum_{n=1}^\infty e^{-4n\frac{\pi}{a}} -1] = \frac{\pi}{a}[\frac{2}{1-e^{-4\frac{\pi}{a}}} -1] =\frac{\pi}{a} coth \frac{2\pi}{a}

    So where did I go wrong to get the 2 instead of a 1?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Aug 2011
    Posts
    16

    Re: Poisson summation

    I know coth \frac{\pi}{a}= \frac{1+e^{-2\frac{\pi}{a}}}{1-e^{-2\frac{\pi}{a}}}, but I have a 4 where the 2 is, and I don't know where I went wrong.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Poisson Square Wave (PSW) and Filtered Poisson Process (FP)
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: April 5th 2010, 12:56 PM
  2. summation help
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 20th 2009, 11:52 AM
  3. Poisson Summation Formula Proof
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 30th 2009, 06:57 PM
  4. Poisson summation formula
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 11th 2008, 10:50 PM
  5. Help with summation
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 14th 2008, 09:19 AM

Search Tags


/mathhelpforum @mathhelpforum