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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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