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

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[[Image:LSU_Key.png|25px|link=http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#Special_Function_Relationships]]
[[Image:LSU_Key.png|25px|link=http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions]]
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<math>~Q_\nu^\mu(z)</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\}
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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)  
[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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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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Latest revision as of 21:39, 1 July 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.

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