Eigenvalues of spheroidal wave functions and their branch points for complex values of propagation constants

Eigenvalues of spheroidal wave functions are calculated as functions of complex propagation constants; they are displayed on a complex plane to form ‘eigenvalue charts.’ These charts show that the eigenvalues have many branch points in the complex planes of their arguments. The values of the branch points and the corresponding eigenvalues are then calculated exactly; a range of several points is shown in a table. These values are the universal constants; they are very important, because they are the basis for determining a complete set of spheroidal wave functions, they give the circle of convergence of the power-series expansions of the eigenvalues, and they also aid in the discovery of domains in which the eigenvalues are obtained by either prolate or oblate asymptotic expansions.