> Have you tried expressing this problem in spherical coodinates?
> There are formulas expressing functions like this in expansions of
> Legendre polynomials and Bessel functions. For example, see
> Jeffreys & Jeffreys, Mathematical methods in Physics.
> That said, if you transform x1 to the origin, you may be able to
> rationalize one of the square roots, and transform it to the standard
> elliptical integral form, see Abramowitz and Stegun.
> Another choice is to try to solv the diff equation using seperation of
> variables, and get a series expansion.
>
Interesting concept to use spherical coordinates for this integration. The
idea for the use of a cubic spline curve to represent part of a magnet set
came out of my efforts back in the 1980s to design high-heat-flux components
(limiters and divertors) for research tokamaks. My idea was to be able to
define the D-shaped magnet sets that represent toroidal field coils more
easily and accurately. So I guess if I were to switch coordinates, I would
want to go to a toroidal coordinate system rather than spherical.
The engineering use of cubic spline curves with which I am familiar is based
entirely on plain and simple cartesian coordinates. Maybe I need to be a bit
more flexible in my thinking.
Thanks for the idea. I'll add it to the list of things to try. Of course
with Legendre polynomials and Bessel functions instead of elliptic
integrals, we're just switching from one set of numerical approximations to
functions to another set. I'll keep my copy of Abramowitz and Stegun handy.
Thanks.
Joe Koski