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Re: Gaussian sums (Was: Speeding up simple Mathematica expressions?)
In article <dogju0$pnl$1 at smc.vnet.net>, Paul Abbott <paul at physics.uwa.edu.au> 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 > > http://functions.wolfram.com/09.03.16.0001.01 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.