Calculation of radial prolate spheroidal wave functions of the second kind

Abstract

Spheroidal wave functions are important for boundary-value calculations in electromagnetics, acoustics and quantum mechanics. This paper discusses the calculation of radial prolate spheroidal wave functions of the second kind for integral mode numbers and real spheroidal parameter. The calculation of these functions is difficult. The existing methods are variously limited by numerical cancellation, slow convergence, algebraic complexity and restricted scope. The paper proposes a hybrid scheme. It uses an analytical solution to give the initial conditions for a numerical solution of the defining differential equation using the Bulirsch–Stoer method. Multiple-precision arithmetic is used to overcome the problem of numerical cancellation. The Wronskian is used to assess the accuracy of the solution. The results are highly accurate. The scheme should provide a practical approach for many applications. Reference results are presented for validation purposes.