       Re: Gaussian sums (Was: Speeding up simple Mathematica expressions?)

• To: mathgroup at smc.vnet.net
• Subject: [mg63307] Re: Gaussian sums (Was: Speeding up simple Mathematica expressions?)
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Fri, 23 Dec 2005 05:08:31 -0500 (EST)
• Organization: The University of Western Australia
• References: <do8ioc\$rvd\$1@smc.vnet.net> <dodec4\$65o\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <dodec4\$65o\$1 at smc.vnet.net>, AES <siegman at stanford.edu>
wrote:

> I originally wrote
>
> >  I'm evaluating a series of functions of which a typical example is
> >
> >  f[a_, x_] := Sum[
> >  Exp[-(Pi a)^2 n^2 -
> >  ((x - n Sqrt[1 - (Pi^2 a^4)])/a)^2],
> >  {n, -Infinity, Infinity}];
>
> and Paul Abbott <paul at physics.uwa.edu.au>   replied:
>
> >  The Mathematica Journal 9(4) under the heading "Sum of Gaussians".
> >  After completing the square of the exponent, such sums can be
> >  expressed in closed form in terms of EllipticTheta functions. See
> >
> >  http://functions.wolfram.com/09.03.06.0001.01

Actually, that is a link to the definition of EllipticTheta[3,z]. I
actually meant to post the link to

http://functions.wolfram.com/09.03.06.0019.01

> >  For this particular example,
> >
> >  f[a_, x_] = a Sqrt[Pi] E^(-(Pi a x)^2) *
> >  EllipticTheta[3, -Pi Sqrt[1 - a^4 Pi^2] x, E^(-(Pi a)^2)]
>
> Thanks for this, and for other replies which pointed me to NSum (now if
> I just had any idea of what the EllipticTheta function is, or does!).
>
> In any event, it's possible some of those who replied would be
> interested in where the interest in this topic came from.  There's a
> coupled fiber optics concept of current practical interest which leads
> to three such sums, namely
>
> f[a_, x_] = Sum[
> Exp[ -(Pi a)^2 n^2 - ((x - n Sqrt[1 - (Pi^2 a^4)])/a)^2],
> {n, -Infinity, Infinity}];
>
> g[a_, x_] = Sqrt[ Pi a^2] Sum[
> Exp[ -(Pi a)^2 n^2 - (Pi a)^2 x^2 - I 2 Pi n Sqrt[1 - (Pi^2 a^4)] x],
> {n, -Infinity, Infinity}];
>
> h[a_, x_] := Sqrt[ Pi a^2] Sum[
> Exp[ -(n/a)^2 - Pi^2 a^2 (x - (I n Sqrt[1 - (Pi^2 a^4)])/(Pi a^2))^2],
> {n, -Infinity, Infinity}];
>
> with the properties that
>
> 0)  The functions g and h are obviously the same except for completing
> the square in the exponent;

Actually, the summands are still different after completing the square
-- symmetry of the summand under n -> -n implies symmetry under x -> -x.

> 1)  The function pairs f and g or h are Fourier transforms of each
> other; and
>
> 2)  All three functions are also *identical* (but with no obvious way to
> convert f into g or h by algebraic methods).
>
> Ergo, f and g or h are apparently a new (or at least not widely
> recognized) family of self-Fourier-transforming functions; the
> references are:
>
>    C. J. Corcoran and K. A. Pasch, "Self-Fourier functions and
> coherent laser combination," J. Phys. A, vol. 37, pp. L461--L469,
> (2004).
>
>    C. J. Corcoran and F. Durville, "Experimental demonstration of a
> phase-locked laser array using a self-Fourier cavity," Appl. Phys.
> Lett., vol. 86, pp. 201118,  (16 May 2005).

I note that there is a small entry on this in Section 1.8 of the
Symbolics Volume of Michael Trott's Mathematica Guidebook.

> A colleague Adnah Kostenbauder has pointed out that this seems to be a
> version of "Jacobi's imaginary transformation" given in Section 21.51 of
> Whittaker and Watson.  Presumably it also has a connection to some
> obscure property of the EllipticTheta functions.

Not an obscure property. It is a basic transformation. See

http://functions.wolfram.com/09.03.16.0001.01

> In physical terms f corresponds an array of narrow, parallel,
> transversely but equally displaced gaussian beams with gaussianly
> decreasing amplitude across the array; g represents a set of wider,
> increasingly tilted gaussian beams all convering onto a common spot; and
> h has the appearance of an array of wide, nominally parallel gaussian
> beams with equal but imaginary-valued transverse displacements.

Most interesting.

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)
AUSTRALIA                               http://physics.uwa.edu.au/~paul

```

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