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MathGroup Archive 2000

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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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