Re: Sum of Gaussian functions

In article <1113209095.578553.230530@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<astanoff@xxxxxxxx> wrote:
>James,

I'm not James but permit me to answer.

>You prove that the sum of gaussians over all integers can be computed
>this way :
>
>s = Sqrt[2*Pi] + Sqrt[2*Pi] +
>Sum[2*Exp[-2*Pi^2*k^2]*Sqrt[2*Pi]*Cos[2*Pi*k*z], {k, 1, oo}]
>
>but, does it really prove that :
>
>s = Sqrt[2*Pi] + Cos[2*Pi*z]*(EllipticTheta[3, 0, 1/Sqrt[E]] -
>Sqrt[2*Pi])
>
>is false ?

Yes, by uniqueness of Fourier coefficients.

Just to double-check I computed values of the various functions
at z=1/3, out to about 300 decimal points. James's sum agrees to
the last decimal place with the original sum. (One only needs a
few dozen summands of the original sum and half a dozen of his summands
to get this kind of accuracy.) The other expression you quote
consists only of the zeroth and first term of his series, and so
differs by roughly the next term of James's series (about 10^{-33}).

Of course, as he points out, his expansion may be proven by
evaluating the Fourier coefficients by integrating. (I had thought
of that but for some reason thought the integrals would be too complicated.)

I'm chiming in here to respond again in this thread because I think
it's such a pleasant series. One can of course create analytic functions
which are periodic or doubly periodic by computing sum F(z-z_i) where
the z_i range over a 1- or 2-dimensional lattice in the plane; that's
how one creates the Weierstrass Pe-function for example. I don't know
why it never occurred to me to try this with something like
F(z) = exp( -z^2 ), for which the convergence of the sum is easy to prove.
Of course for a periodic function one has a Fourier expansion, and the
rapid decay of this F makes for a rapid decay of the Fourier coefficients.
(The first few are on the order of 10^-7, 10^-33, 10^-76, 10^-136, ...)
So numerically it takes some work to see that the function is not
constant, and more work to see that the function is not a simple cosine.

As it happens, I was tidying up my files yesterday and came across
a recent sci.math thread asking whether the series sum( sin(n^2)/n )
converges. I wonder if that can be related to this series?
There are superficial resemblances.

dave
.

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