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 (Gil et al., 2012) for the computation of conical functions. Specifically, the module includes now a routine for computing the function ${{\rm R}}^{m}_{-\frac{1}{2}+i\tau}(x)$, 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.

The module also improves the performance of our previous algorithm for the conical function ${\rm P}^m_{-\tfrac12+i\tau}(x)$ and it includes now the computation of the first order derivative of the function. This is also considered for the function ${{\rm R}}^{m}_{-\frac{1}{2}+i\tau}(x)$ in the extended algorithm.