Difference between revisions of "User:Tohline/Math/EQ Toroidal04"
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<math>~ | <math>~(\nu - \mu + 1)P^\mu_{\vu + 1}</math> | ||
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[https:// | [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 & Stegun (1995)], p. 334, eq. (8.5.3) | ||
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Revision as of 23:47, 13 June 2018
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<math>~(\nu - \mu + 1)P^\mu_{\vu + 1}</math> |
<math>~=</math> |
<math>~ 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\} </math> |
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Abramowitz & Stegun (1995), p. 334, eq. (8.5.3) |
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NOTE: Both <math>~P_\nu^\mu</math> and <math>~Q_\nu^\mu</math> satisfy this same recurrence relation. |