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_ Sent via Deja.com http://www.deja.com/ Share what you know. Learn what you don't. ============================================================================== 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_ Sent via Deja.com http://www.deja.com/ Before you buy.