Calculating the branch points of the eigenvalues of the Coulomb spheroidal wave equation

A method for computing the eigenvalues $\lambda_{mn}(b,c)$ and the eigenfunctions of the Coulomb spheroidal wave equation is proposed in the case of complex parameters $b$ and $c$. The solution is represented as a combination of power series expansions that are then matched at a single point. An extensive numerical analysis shows that certain $b_s$ and $c_s$ are second-order branch points for $\lambda_{mn}(b,c)$ with different indices $n_1$ and $n_2$, so that the eigenvalues at these points are double. Padé approximants, quadratic Hermite–Padé approximants, the finite element method, and the generalized Newton method are used to compute the branch points $b_s$ and $c_s$ and the double eigenvalues to high accuracy. A large number of these singular points are calculated.