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

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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