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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)

Valid for:    

~\mathrm{Re} ~\nu > -\tfrac{1}{2} 

    and    

~\mathrm{Re} (\nu + \mu + 1) > 0 \, .

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