Calculation of spheroidal wave functions

This paper discusses the calculation of prolate spheroidal wave functions for integral mode numbers and real spheroidal parameter. The functions have applications in electromagnetism, acoustics, quantum mechanics and finite Fourier analysis. The objective is the use of large values of mode number and spheroidal parameter. This capability allows the solution of the Helmholtz equation when the spheroid is large compared with the wave-length. The paper develops the theory, describes demonstration software and gives several numerical examples. Multiple-precision calculations are also explored. For many purposes the calculation of the eigenvalues and the regular angular function can be regarded as completely solved. Accurate results may be obtained for the radial functions, but the scope is limited by numerical cancellation and slow convergence.