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
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- From: Andrzej Kozlowski <akoz@mimuw.edu.pl>
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- Infinite sum of gaussians
- From: "Valeri Astanoff" <astanoff@yahoo.fr>
- Re: Infinite sum of gaussians
- From: "Valeri Astanoff" <astanoff@yahoo.fr>
- Infinite sum of gaussians