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

From VistrailsWiki
Jump to navigation Jump to search
 
(One intermediate revision by the same user not shown)
Line 3: Line 3:
<tr>
<tr>
<td align="right">
<td align="right">
[[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]]
</td>  
</td>  
   <td align="right">
   <td align="right">
<math>~(\nu - \mu + 1)P^\mu_{\vu + 1}</math>
<math>~(\nu - \mu + 1)P^\mu_{\nu + 1} (z)</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 13: Line 13:
   <td align="left">
   <td align="left">
<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>
   </td>
   </td>
Line 30: Line 27:
<tr>
<tr>
   <td align="right">
   <td align="right">
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.
   </td>
   </td>
</tr>
</tr>
</table>
</table>

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.