       Re: Infinite sum of gaussians

• To: mathgroup at smc.vnet.net
• Subject: [mg56055] Re: Infinite sum of gaussians
• From: "Carl K. Woll" <carlw at u.washington.edu>
• Date: Thu, 14 Apr 2005 08:55:25 -0400 (EDT)
• Organization: University of Washington
• References: <200504120926.FAA27573@smc.vnet.net> <d3ibr6\$9un\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```"Andrzej Kozlowski" <akoz at mimuw.edu.pl> wrote in message
news:d3ibr6\$9un\$1 at smc.vnet.net...
> On 12 Apr 2005, at 18:26, Valeri Astanoff 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])
>>
[snip]
>
> Here is a sketch of an attempted proof. It is only a sketch (I do not
> have much free time to spend on this) and I have not checked all the
> details but I am pretty sure it can be made rigorous..
>
[snip]
>
> But since the function is analytic everywhere it's values are
> determined by it's value and the values of all its derivatives at just
> a single point. So it should be 2Pi everywhere.
>

But, why do you suppose the function is analytic?

At any rate, if you plug in something like z=1/2, and evaluate both sides to
high precision, say 100 digits, you see that Mathemtaica believes the
equation is not true. In fact, it's not too difficult to find the Fourier
expansion of the left hand side, and one finds that the Fourier series does
not end after 2 terms, but rather has an infinite number of terms.

Carl Woll

> Andrzej Kozlowski
> Chiba, Japan
> http://www.akikoz.net/andrzej/index.html
> http://www.mimuw.edu.pl/~akoz/
>

```

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