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

A method for calculating eigenvalues λmn(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 cs are the branch points of the functions λmn(c) with different indexes n1 and n2 so that the value λmn1 (cs) is a double one: λmn1 (cs) = λmn2 (cs). The numerical analysis suggests that, for each fixed m, all the branches of the eigenvalues λmn(c) corresponding to the even spheroidal functions form a complete analytic function of the complex argument c. Similarly, all the branches of the eigenvalues λmn(c) corresponding to the odd spheroidal functions form a complete analytic function of c. To perform highly accurate calculations of the branch points cs of the double eigenvalues λmn(cs), the Padé approximants, the Hermite-Padé quadratic approximants, and the generalized Newton iterative method are used. A large number of branch points are calculated.