#### Computation of a numerically satisfactory pair of solutions of the differential equation for conical functions of non-negative integer orders

##### T. M. Dunster, A. Gil, J. Segura, N. M. Temme

We consider the problem of computing satisfactory pairs 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 ${Q}^{\mu}_{-\frac12+i\tau}(x)$, $x>-1$. An algorithm for computing a numerically satisfactory pair of solutions is already available when $-11$, the function $\Re\left\{e^{-i\pi \mu} {{Q}}^{\mu}_{-\frac{1}{2}+i\tau}(x) \right\}$. The proposed algorithm allows the computation of the function on a large parameter domain without requiring the use of extended precision arithmetic.

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