Evaluation of eigenvalues and eigenfunctions of Coulomb spheroidal wave equation

A method for evaluation the eigenvalues $\lambda_{m,q}(b,c)$ and the eigenfunctions of Coulomb spheroidal wave equation in a case of complex parameters $b$ and $c$ is proposed. The method is based on construction of two expansions of solution at the singular points $\eta=\pm1$ and on matching of the expansions at the point $\eta=0$. Numerical experiments show that there exist branching points of the eigenvalues $\lambda_{m,q}(b,c)$ for some complex values $b$ or $c$, the order of the branching points is equal $2$.