Re: A hard serie : sum_n exp^(-a*n²)



In article
<b8b2131b-d2e8-4fa4-a5f2-65c175f26cae@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
ettevy <han.wang1981@xxxxxxxxx> wrote:

Hello,
Does anyone know how to calculate the following power serie:
f(a)=sum_n exp(-a*n*n), with a>0 a real number and n natural number
from 0 to +infty.

I need the analytic expression of this serie. It seems to be a
approximation of the gaussian integral which is easy to calculate.

Thanks

Ettevy

It is a theta function...

In Maple notation,
JacobiTheta3(z,q)
= 1+2*sum(q^(n^2)*cos((2*n)*z), n = 1 .. infinity);
so
(JacobiTheta3(0, exp(-a))+1)/2
= 1+sum(exp(-n^2*a), n = 1 .. infinity)

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
.