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<math>~Q_\nu^\mu(z)</math>
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<math>~(\nu - \mu + 1)P^\mu_{\vu + 1}</math>
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[https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erd&eacute;lyi (1953)]:&nbsp; Volume I, &sect;3.7, p. 156, eq. (10)  
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[https://books.google.com/books?id=MtU8uP7XMvoC&printsec=frontcover&dq=Abramowitz+and+stegun&hl=en&sa=X&ved=0ahUKEwialra5xNbaAhWKna0KHcLAASAQ6AEILDAA#v=onepage&q=Abramowitz%20and%20stegun&f=false Abramowitz &amp; Stegun (1995)], p. 334, eq. (8.5.3)
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Revision as of 16:47, 13 June 2018

Failed to parse (unknown function\vu): ~(\nu - \mu + 1)P^\mu_{\vu + 1}

~=

~
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\}

Abramowitz & Stegun (1995), p. 334, eq. (8.5.3)

NOTE: Both ~P_\nu^\mu and ~Q_\nu^\mu satisfy this same recurrence relation.

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