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MathGroup Archive 2005

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Re: Re: Infinite sum of gaussians


Valeri Astanoff wrote:
> Andrzej,
> 
> I'm not a mathematician, just an engineer, and what puzzles me is this
> :
> 
> In[1]:=
> { Sum[Exp[-(z-k)^2/2], {k, -Infinity, Infinity}],
>       Sqrt[2*Pi]+
>         Cos[2*Pi*z]*(EllipticTheta[3,0,1/Sqrt[E]]-Sqrt[2*Pi])} /. z ->
> 1/2 //
>   N[#,35]&
> 
> Out[1]=
> {2.5066282612190954600008515157581306,
>  2.5066282612190954600008515157581301}
> 
> If I trust mathematica, this should induce to think
> the equality is false.
> 
> Anyway, thanks for the time you spent.
> 
> 
> Regards,
> 
> v.a.

If I am seeing those numbers correctly, they differ only in the last 
digit. So I would be strongly induced to think the equality is true (at 
least for z=1/2), and that the numeric evaluation of
Sum[E^(-(1/2-k)^2/2), {k,-Infinity,Infinity}]
at 35 digits has done something bordering on magic.

By the way, here is an identity. That sum of Gaussians is

Exp[-z^2/2]*EllipticTheta[3,z/(2*I),1/Sqrt[E]]

This follows from

http://functions.wolfram.com/EllipticFunctions/EllipticTheta3/06/01/

using the second q-series formula.


Daniel Lichtblau
Wolfram Research


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