Computing the Conical Function $P^{\mu}_{-1/2+i\tau}(x)$
Amparo Gil, Javier Segura, Nico Μ. Τemme · SIAM Journal on Scientific Computing · 2009
A stable computational scheme for the conical function $P^{\mu}_{-1/2+i\tau}(x)$ for $x>-1$, real $\tau$, and $\mu\le 0$ or $\mu\in\mathbb{N}$ is presented. The scheme combines uniform asymptotic expansions for large $|\mu|$ with the application of the three-term recurrence relation on the $\mu$ index in the direction of decreasing $|\mu|$ when $x>0$. When $x<0$, the conditioning of recursion is the opposite, and conical functions can be computed in the direction of increasing $|\mu|$.