Absolute and uniform convergence of alternate forms of the prolate spheroidal radial wave functions

Abstract

A new orthonormal basis set representation of the prolate spheroidal radial and angular wave functions is presented. The embedded series solutions to a fully-coupled fluid-solid interaction continuum physics problem is defined by product sets of Legendre polynomials and modified spherical Bessel functions of the first and third kinds. We prove that the embedded series solutions analytically converge absolutely and uniformly to the exact solutions of the system of coupled continuum equations. The satisfaction of the bilinear concomitant and its utility in establishing the convergence proofs is demonstrated.