  A previous article showed that alternative expressions for calculating oblate
spheroidal radial functions of both kinds can provide accurate values over very
large parameter ranges using double precision arithmetic, even where the
traditional expressions fail. The size parameter c was assumed real. This paper
considers the case where c = cr + ici is complex with an imaginary part ci
often used to represent losses in wave behavior. The methods for c real
modified to complex arithmetic work reasonably well as long as ci is very
small. This paper describes the substantial changes necessary to obtain useful
results for larger values of ci. It shows that accurate eigenvalues can usually
be obtained even though the matrix methods used to obtain them for c real
provide increasingly inaccurate values, primarily for those with relatively
small magnitude, as ci increases. It also shows that some of the eigenvalues
can be prolate-like with values that are well approximated using asymptotic
estimates for prolate eigenvalues where c is replaced with -ic. A method to
order the eigenvalues is presented. The modifications necessary to compute
accurately the radial and angular functions for complex c are discussed. A
resulting Fortran program coblfcn provides useful function values for a
reasonably wide range of c, the order m, and the radial coordinate when using
double precision arithmetic. The results can be improved by using quadruple
precision for the Bouwkamp procedure to ensure accurate double precision
eigenvalues. Further improvement is obtained using full quadruple precision.
Coblfcn is freely available at www.mathieuandspheroidalwavefunctions.com
