COMPUTING THE CONICAL FUNCTION P-1/2+iτμ (x)

A stable computational scheme for the conical function P--1/2+i tau(mu) (x) for x > -1, real tau, and mu <= 0 or mu is an element of N is presented. The scheme combines uniform asymptotic expansions for large vertical bar mu vertical bar with the application of the three-term recurrence relation on the mu index in the direction of decreasing vertical bar mu vertical bar when x > 0. When x < 0, the conditioning of recursion is the opposite, and conical functions can be computed in the direction of increasing vertical bar mu vertical bar.