Approximations to toroidal harmonics

Toroidal harmonics P/sub n-1/2/1(cosh μ) and Q/sub n-1/2/1(cosh μ) 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 < μ < infinity. The functional form used for these approximations is dictated by their behavior as μ → 0 and as μ → infinity, 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 < μ < 5, and in the best cases these peaks are less than 10-10. The simple method used to determine the approximations is described. Relative error curves are also presented, obtained by comparing approximations to the more accurate values computed by direct summation of the hypergeometric series