# User:Tohline/Math/EQ Toroidal04

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 $~Q_\nu^\mu(z)$ $~=$ $~ e^{i \mu \pi} ~ (2\pi)^{-\frac{1}{2}} (z^2-1)^{\mu/2} ~\Gamma(\mu + \tfrac{1}{2})~\biggl\{ \int_0^\pi (z - \cos t)^{-\mu - \frac{1}{2}} \cos[(\nu + \tfrac{1}{2})t] ~dt -\cos(\nu\pi) \int_0^\infty (z + \cosh t)^{-\mu - \frac{1}{2}} e^{-(\nu + \frac{1}{2})t} ~dt \biggr\}$ A. Erdélyi (1953):  Volume I, §3.7, p. 156, eq. (10)
 NOTE: Both $~P_\nu^\mu$ and $~Q_\nu^\mu$ satisfy this same recurrence relation.