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

A method for calculating eigenvalues λ_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 λ_mn(c) with different indexes n ₁ and n ₂ so that the value λ_{mn 1} (c _s ) is a double one: λ_{mn 1} (c _s ) = λ_{mn 2} (c _s ). The numerical analysis suggests that, for each fixed m, all the branches of the eigenvalues λ_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 λ_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 λ_mn(c _s), the Padé approximants, the Hermite-Padé quadratic approximants, and the generalized Newton iterative method are used. A large number of branch points are calculated.