Difference between revisions of "User:Tohline/Math/EQ Toroidal04"

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<math>~(\nu - \mu + 1)P^\mu_{\vu + 1}</math>
<math>~(\nu - \mu + 1)P^\mu_{\nu + 1} (z)</math>
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<math>~
<math>~
e^{i \mu \pi} ~ (2\pi)^{-\frac{1}{2}} (z^2-1)^{\mu/2} ~\Gamma(\mu + \tfrac{1}{2})~\biggl\{
(2\nu + 1)z P_\nu^\mu(z) - (\nu + \mu)P^\mu_{\nu-1}(z)
\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>
</math>
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NOTE:  Both <math>~P_\nu^\mu</math> and <math>~Q_\nu^\mu</math> satisfy this same recurrence relation.
NOTE:  <math>~Q_\nu^\mu</math>, as well as <math>~P_\nu^\mu</math>, satisfies this same recurrence relation.
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Revision as of 23:53, 13 June 2018

LSU Key.png

<math>~(\nu - \mu + 1)P^\mu_{\nu + 1} (z)</math>

<math>~=</math>

<math>~ (2\nu + 1)z P_\nu^\mu(z) - (\nu + \mu)P^\mu_{\nu-1}(z) </math>

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

NOTE: <math>~Q_\nu^\mu</math>, as well as <math>~P_\nu^\mu</math>, satisfies this same recurrence relation.