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

## TL;DR

This paper introduces an extended computational module for conical functions, enabling accurate calculation of a new companion function crucial for solving boundary value problems in conical domains.

## Contribution

The paper presents an extension to the CONICAL module, adding routines to compute a new numerically satisfactory conical function for improved problem-solving.

## Key findings

- Provides a routine for computing ${R}^{m}_{-rac{1}{2}+i au}(x)$
- Enables solving Dirichlet problems in conical domains
- Enhances the computational tools for conical functions

## Abstract

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 ${{\rm R}}^{m}_{-\frac{1}{2}+i\tau}(x)$, 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.