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: > > [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). 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