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Re: Infinite sum of n^2 Exp[-n^2]
*To*: mathgroup at smc.vnet.net
*Subject*: [mg26486] Re: [mg26477] Infinite sum of n^2 Exp[-n^2]
*From*: BobHanlon at aol.com
*Date*: Thu, 28 Dec 2000 02:52:24 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
This may not be what you want, but it may help you after some investigation
of EllipticTheta.
Let s equal
Sum[n^2 * Exp[-a* n^2], {n, -Infinity, Infinity}];
Since
n^2 * Exp[-a * n^2] == -D[Exp[-a * n^2], a]
True
then for
s1 = Sum[Exp[-a* n^2], {n, -Infinity, Infinity}]
EllipticTheta[3, 0, E^(-a)]
we have
s = -D[s1, a]
Derivative[0, 0, 1][EllipticTheta][3, 0, E^(-a)]/E^a
Plot[s, {a, 1, 5}];
est = Sum[n^2 * Exp[-a* n^2], {n, -25, 25}];
Plot[est, {a, 1, 5}];
Table[s - est, {a, 1., 5., .1}] // Chop
{-7.57002049933675*^-9, -2.2411962286028597*^-9,
-8.026793674176247*^-10, -3.2794589266416097*^-10,
-1.4620038513157851*^-10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0}
Bob Hanlon
In a message dated 12/22/00 11:08:44 PM, siegman at stanford.edu writes:
>Mathematica can do the infinite sums (-Infinity to Infinity) of
>
> Exp[-n^2]
>
>and also
>
> Exp[- a n^2]
>
>in closed form, but not
>
> n^2 Exp[-n^2]
>
>or better
>
> n^2 Exp[- a n^2]
>
>Are there known closed analytical forms for these?
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