Conical: an extended module for computing a numerically satisfactory pair of solutions of the differential equation for conical functions

Conical functions appear in a large number of applications in physics and engineering. In this paper we describe an extension of our module CONICAL for the computation of conical functions. Specifically, the module includes now a routine for computing the function ${\mathrm{R}}_{-\frac{1}{2}+i\tau }^m(x)$, a real-valued numerically satisfactory companion of the function ${\rm P}^m_{-\tfrac12+i\tau}(x)$for$x>1$. In this way, a natural basis for solving Dirichlet problems bounded by conical domains is provided.