[Date Index] [Thread Index] [Author Index]

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

• To: mathgroup at smc.vnet.net
• Subject: [mg63330] Re: Gaussian sums (Was: Speeding up simple Mathematica expressions?)
• From: AES <siegman at stanford.edu>
• Date: Sat, 24 Dec 2005 07:18:55 -0500 (EST)
• Organization: Stanford University
• References: <do8ioc$rvd$1@smc.vnet.net> <dodec4$65o$1@smc.vnet.net> <dogju0$pnl$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

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.