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Gaussian sums (Was: Speeding up simple Mathematica expressions?)
*To*: mathgroup at smc.vnet.net
*Subject*: [mg63279] Gaussian sums (Was: Speeding up simple Mathematica expressions?)
*From*: AES <siegman at stanford.edu>
*Date*: Thu, 22 Dec 2005 00:04:42 -0500 (EST)
*Organization*: Stanford University
*References*: <do8ioc$rvd$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
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
>
> 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;
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:
[1] C. J. Corcoran and K. A. Pasch, "Self-Fourier functions and
coherent laser combination," J. Phys. A, vol. 37, pp. L461--L469,
(2004).
[2] 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).
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.
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.
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