From: Robin Chapman
Subject: Re: Sum involving exp
Date: Mon, 14 Jun 1999 12:54:43 GMT
Newsgroups: sci.math
Keywords: theta-functions
In article ,
qqquet@hotbot.com (Leroy Quet) wrote:
> I'm sure this has been asked before,but
> is there a closed expression for the following?
>
> sum_{m=0}^infinity [exp(-xm^2)],
>
> for positive x.
>
This is essentially a theta-function. One (of several) definitions
of theta(z) is
theta(z) = sum_{m=-infinty}^\infinity exp(pi i m^2 z).
With this definition, the sum in question is
(theta(x i/pi) + 1)/2.
--
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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==============================================================================
From: Robin Chapman
Subject: Re: Discrete Gaussian Sums
Date: Wed, 20 Oct 1999 12:59:45 GMT
Newsgroups: sci.math
In article <380D045E.2E95@yahoo.com>,
Yves Capdeboscq wrote:
> Hello,
>
> Probably a textbook question, but I haven't found it (maybe because I
> did not know excatly where to look)
>
> What is
>
> sum_{- infty}^{+ infty} exp(-n*n/2) - exp(-(n+1/2)*(n+1/2)/2
>
> Or even better, what is
>
> sum_{- infty}^{+ infty} exp(-A*n*n/2) - exp(-A*(n+1/2)*(n+1/2)/2
>
> given a positive constant A?
>
> If it is indeed a textbook question, would you have a reference?
These are theta functions. Two of the classical theta
functions are
theta_3(tau) = sum_{n=-infinity}^infinity q^{n^2}
and
theta_2(tau) = sum_{n=-infinity}^infinity q^{(n + 1/2)^2}
where tau is in the upper half plane and q = exp(pi i tau).
Taking tau = ti will give these sums for suitable t.
Books:
Whittaker & Watson, A Course of Modern Analysis,
Rademacher's book on analytic number theory,
McKean & Moll, Elliptic Curves
--
Robin Chapman
http://www.maths.ex.ac.uk/~rjc/rjc.html
"`Well, I'd already done a PhD in X-Files Theory at UCLA, ...'"
Greg Egan, _Teranesia_
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