Accurate Positions of Branch Points of Minimum Magnitudes and Their Associated Spheroidal Eigenvalues

The Newton- Raphson method with two complex variable is utilized to compute the branch points with minimum magnitudes and their associated eigenvalues in the first quadrant of the complex plane with the parameter c =kF, where k is a complex wave number, and F is the semifocal length of the spheroidal system. The efficient numerical method which is applied to the spheroidal eigenvalue equation and the equation of its partial derivative with respect to the eigenvalue, is used to simultaneously solve the two complex variables, the branch point and its associated eigenvalue with a high precision, from which they can be tabulated for references.