Re: Re: Infinite sum of gaussians

*To*: mathgroup at smc.vnet.net*Subject*: [mg56077] Re: [mg56032] Re: Infinite sum of gaussians*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Fri, 15 Apr 2005 04:47:10 -0400 (EDT)*References*: <200504120926.FAA27573@smc.vnet.net><d3ibr6$9un$1@smc.vnet.net> <200504141254.IAA28085@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

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

**Follow-Ups**:**Re: Re: Re: Infinite sum of gaussians***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**References**:**Infinite sum of gaussians***From:*"Valeri Astanoff" <astanoff@yahoo.fr>

**Re: Infinite sum of gaussians***From:*"Valeri Astanoff" <astanoff@yahoo.fr>