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In this paper we describe an algorithm and a Fortran 90 module ({\bfConical}) for the computation of the conical function$P^m_{-\tfrac12+i\tau}(x)$ for $x>-1$, $m \ge 0$, $\tau >0$. These functionsappear in the solution of Dirichlet problems for domains bounded by cones;because of this, they are involved in a large number of applications inEngineering and Physics.  In the Fortran 90 module, the admissible parameter ranges for computing theconical functions in standard IEEE double precision arithmetic are restrictedto $(x,m,\tau) \in (-1,1) \times [0,\,40] \times [0,\,100]$ and $(x,m,\tau) \in(1,100) \times [0,\,100] \times [0,\,100]$. Based on tests of the three-termrecurrence relation satisfied by these functions and direct comparison withMaple, we claim a relative accuracy close to $10^{-12}$ in the full parameterrange, although a mild loss of accuracy can be found at some points of theoscillatory region of the conical functions. The relative accuracy increases to$10^{-13}\,-\,10^{-14}$ in the region of the monotonic regime of the functionswhere integral representations are computed ($-1<x<0$).

An improved algorithm and a Fortran 90 module for computing the conical function $P^m_{-1/2+i\tau}(x)$