Calculation of the branch points of the eigenfunctions corresponding to wave spheroidal functions

A method for calculating eigenvalues $\lambda_{mn}(c)$ corresponding to the wave spheroidal functions in the case of a complex parameter c is proposed, and a comprehensive numerical analysis is performed. It is shown that some points $c_s$ are the branch points of the functions $\lambda_{mn}(c)$ with different indexes $n_1$ and $n_2$ so that the value $\lambda_{mn_1}(c_s)$ is a double one: $\lambda_{mn_1}(c_s)=\lambda_{mn_2}(c_s)$. The numerical analysis suggests that, for each fixed $m$, all the branches of the eigenvalues $\lambda_{mn}(c)$ corresponding to the even spheroidal functions form a complete analytic function of the complex argument $c$. Similarly, all the branches of the eigenvalues $\lambda_{mn}(c)$ corresponding to the odd spheroidal functions form a complete analytic function of $c$. To perform highly accurate calculations of the branch points $c_s$ of the double eigenvalues $\lambda_{mn}(c)$, the Padé approximants, the Hermite–Padé quadratic approximants, and the generalized Newton iterative method are used. A large number of branch points are calculated.