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 $-1

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