We consider the problem of computing satisfactory pair of solutions of the differential equation for Legendre functions of non-negative integer order $\mu$ and degree $-\frac12+i\tau$, where $\tau$ is a non-negative real parameter. Solutions of this equation are the conical functions ${\rm{P}}^{\mu}_{-\frac12+i\tau}(x)$ and ${\rm Q}^{\mu}_{-\frac12+i\tau}(x)$, $x>-1$. An algorithm for computing a numerically satisfactory pair of solutions is already available when $-1<x<1$ (see \cite{gil:2009:con}, \cite{gil:2012:cpc}). In this paper, we present a stable computational scheme for a real valued numerically satisfactory companion of the function ${\rm{P}}^{\mu}_{-\frac12+i\tau}(x)$ for $x>1$, the function $\Re\left\{e^{-i\pi \mu} {{\rm Q}}^{\mu}_{-\frac{1}{2}+i\tau}(x) \right\}$. The proposed algorithm allows the computation of the function on a large parameter domain without requiring the use of extended precision arith-metic.

Additional Metadata
Keywords Legendre functions, conical functions, three-term recurrence relations, numerical methods for special functions
MSC Computation of special functions, construction of tables (msc 65D20)
THEME Other (theme 6)
Stakeholder Unspecified
Citation
Dunster, T.M, Gil, A, Segura, J, & Temme, N.M. (2014). Computation of a numerically satisfactory pair of solutions of the differential equation for conical functions of non-negative integer orders.