From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: summation methods
Date: Mon, 07 Jun 1999 13:14:10 GMT
Newsgroups: sci.math
Keywords: Poisson's summation formula

In article <7jgc33$lp1$1@nnrp1.deja.com>,
  Serge Zlobin <sirg@my-deja.com> wrote:
>  In a certain article I've read that the infinite sum (-1)^n/(n^2+1) can
> be evaluate through Poisson's summation formula or Mittag-Leffler
> expansions of trigonometric functions. Can someone explain what these
> methods are?

Poisson summation:

sum_{n=-infinfy}^infinity f(n) = sum_{m=-infinfy}^infinity F(m)

where F is the Fourier transform of f:
F(t) = integral_{-infinity}^infinity exp(- 2 pi i xt) f(x) dx,

provided that f satisfies some mild analytic conditions: for instance
that it be differentiable and it and its derivative decay resonably
rapidly at infinity.

--
Robin Chapman
 http://www.maths.ex.ac.uk/~rjc/rjc.html
  "They did not have proper palms at home in Exeter."
    Peter Carey, _Oscar and Lucinda_


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