       Re: Summations

• To: mathgroup at smc.vnet.net
• Subject: [mg26576] Re: Summations
• From: BobHanlon at aol.com
• Date: Tue, 9 Jan 2001 01:51:50 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

This may or may not extend the evaluation far enough to meet your needs. If
it doesn't, it may suggest an approach to look for an alternate expression
which meets your requirement.

Needs["Graphics`Graphics`"];

Clear[s1, s2, s3];

Looking for Sum[n^2 * E^(-a*n^2), {n, -Infinity, Infinity}]

n^2 * E^(-a*n^2) \[Equal] -D[E^(-a*n^2), a]

True

s1[a_] := Evaluate[Sum[E^(-a*n^2), {n, -Infinity, Infinity}]]

s2[a_] :=  Evaluate[-D[s1[a], a]]

The expression for s2 does not evaluate numerically for values of a which are
less than about 0.5466.  However, using the relation between EllipticTheta
and EllipticK given in Gradshteyn & Ryzhik (8.197.1), an alternate expression
(s3) which evaluates numerically for lower values of a can be found.

s3[a_] := Evaluate[
Simplify[-D[Sqrt[2*EllipticK[InverseEllipticNomeQ[E^-a]]/Pi], a]]]

DisplayTogether[
Plot[s2[a], {a, 0.5466, 2.},
PlotStyle -> {AbsoluteDashing[{3,5}], RGBColor[0, 0, 1]}],
Plot[s3[a], {a, 0.266, 2.},
PlotStyle -> {AbsoluteDashing[{5,3}], RGBColor[1, 0, 0]}],
PlotRange -> All, Axes -> False, Frame -> True];

Bob Hanlon

In a message dated 2001/1/2 11:16:56 AM, siegman at stanford.edu writes:

>Your responses to my query (McCann's msg appended below) looked at
>first glance like exactly what I needed -- the technique you both
>suggested of using
>
>   f[a_]=Sum[Exp[-a n^2],{n,1,Infinity}]
>
>   f'[a]==-Sum[n^2 Exp[-a n^2],{n,1,Infinity}]
>
>gives an analytical result for the  n^2 Exp[-a n^2] sum that involves
>the Mathematica function EllipticThetaPrime[] (although Mathematica displays it in a
>different format).
>
>As soon as I tried to use this to do some calculations, however, I
>found that although  f[a]  (which involves EllipticTheta[]) plots
>fine and is a smooth real-valued function for any value of  a > 0,  f'[a]
>will only return numerical values for a > 0.5xxx, where 0.5xxx seems
>to be some special value around 0.52 or 0.53.  Unfortunately, the
>problem I'm interested in involves smaller values of  a (.e., wider
>gaussian distributions).
>

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