  In this paper we study the eigenvalues of the angular spheroidal wave
equation and its generalization, the Coulomb spheroidal wave equation. An
associated differential system and a formula for the connection coefficients
between the various Floquet solutions give rise to an entire function whose
zeros are exactly the eigenvalues of the Coulomb spheroidal wave equation. This
entire function can be calculated by means of a recurrence formula with
arbitrary accuracy and low computational cost. Finally, one obtains an
easy-to-use method for computing spheroidal eigenvalues and the corresponding
eigenfunctions.
