APPROXIMATIONS TO TOROIDAL HARMONICS

No. DE-AC02-78ET51013. Reproduction, translation, publication, use and disposal, in whole or in part by or for the United States govern-ment is permitted. Approximations to Toroidal Harmonics Toroidal harmonics P- 1/2 (cosh g) and Q- 1 2 (cosh y) are useful in solutions to Maxwell's equations in toroidal coordinates. In order to speed their computation, a set of approximations has been developed that is valid over the range 0 < g < 00. The functional form used for these approximations is dictated by their behavior as a- 0 and as A- oc, and is similar to that used by Hastings in his approximations to the elliptic integrals K and E. This report lists approximations of several mathematical forms with varying numbers of terms; approximations to the above Legendre functions are given for n = 0 through 6. Coefficients of each expansion have been adjusted to distribute the relative error in equi-amplitude peaks over some range, typically.05 < u < 5, and in the best cases these peaks are less than 10- ' 0. The simple method used to determine the approximations is described. Relative error curves ar