On Calculations of Legendre Functions and Associated Legendre Functions of the First Kind of Complex Degree

Formulas for calculating Legendre functions and associated Legendre functions of the first kind of complex degree using an Ermakov–Lewis invariant are presented. These formulas are straight-forward to implement numerically and are motivated by the lack of computational routines in standard university tools like those of MatLab and Maple. Angular waves propagating in opposite directions are also obtained. The results are particularly useful in complex angular momentum theories and nearside/farside analysis of spin-dependent angular scattering from central potentials.