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

In article <dogju0$pnl$1 at>,
 Paul Abbott <paul at> wrote:

> > 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

   Working definition of "obscure" = "Something I don't know"
   (but that Paul and David Lichtbau generally do)

> > 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.

   I'll post some illustrations of the optical beam embodiments
   on my web page eventually -- but it may take a little while.

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