Title: Accurate calculation of oblate spheroidal wave functions,
Abstract: Alternative expressions for calculating the oblate spheroidal radial
functions of both kinds R1ml and R2ml are shown to provide accurate values over
very large parameter ranges using 64 bit arithmetic, even where the traditional
expressions fail. First is the expansion of the product of a radial function
and the angular function of the first kind in a series of products of the
corresponding spherical functions, with the angular coordinate being a free
parameter. Setting the angular coordinate equal to zero leads to accurate
values for R2ml when the radial coordinate xi is larger than 0.01 and l is
somewhat larger than m. Allowing it to vary with increasing l leads to highly
accurate values for R1ml over all parameter ranges. Next is the calculation of
R2ml as an integral of the product of S1ml and a spherical Neumann function
kernel. This is useful for smaller values of xi. Also used is the near equality
of pairs of low order eigenvalues when the size parameter c is large that leads
to accurate values for R2ml using neighboring accurate values for R1ml. A
modified method is described that provides accurate values for the necessary
expansion coefficients when c is large and l is near m and traditional methods
fail. A resulting Fortran computer program Oblfcn almost always provides radial
function values with at least 8 accurate decimal digits using 64 bit arithmetic
for m up to at least 1000 with c up to at least 2000 when xi is greater than
0.000001 and c up to at least 5000 when xi is greater than 0.01. Use of 128 bit
arithmetic extends the accuracy to 15 or more digits and extends xi to all
values other than zero. Oblfcn is freely available.