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MathGroup Archive 2005

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Re: Infinite sum of gaussians

  • To: mathgroup at smc.vnet.net
  • Subject: [mg56072] Re: Infinite sum of gaussians
  • From: Maxim <ab_def at prontomail.com>
  • Date: Thu, 14 Apr 2005 08:56:45 -0400 (EDT)
  • References: <d3g88u$sdd$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

On Tue, 12 Apr 2005 10:35:10 +0000 (UTC), Valeri Astanoff  
<astanoff at yahoo.fr> wrote:

> Dear group,
>
> Could anyone prove or disprove this equality, at least for z being real
> :
>
> 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])
>
> ?
>
>
> Valeri
>

No, this identity is not true. Here's a numerical check:

In[1]:=
Plot[Module[{z = SetPrecision[zz, 50]},
     NSum[Exp[-(z - k)^2/2], {k, -Infinity, Infinity},
         WorkingPrecision -> 50] -
       (Sqrt[2*Pi] + Cos[2*Pi*z]*
         (EllipticTheta[3, 0, 1/Sqrt[E]] - Sqrt[2*Pi])) ],
   {zz, 0, 1}, Compiled -> False]

In[2]:=
Max@ Abs@ Cases[%, Line[L_] :> L[[All, 2]], -1]

Out[2]=
5.136067375830557*^-34

Even though the values are within 10^-33 from zero, Plot clearly indicates  
that we have a non-zero function. Note that this wouldn't work with  
Compiled -> True: the result of SetPrecision would be immediately  
converted back to machine real during the assignment to z.

Using the formula http://functions.wolfram.com/09.03.06.0019.01 , we can  
express the sum in terms of EllipticTheta as follows:

Sum[Exp[-(z - k)^2/2], {k, -Infinity, Infinity}] ==

Sqrt[2*Pi]*EllipticTheta[3, Pi*z, E^(-2*Pi^2)].

Maxim Rytin
m.r at inbox.ru


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