Template:Math/EQ Toroidal05 - Revision history

Jet53man: Created page with "

Image:LSUkey.png|25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Le..."
2021-07-18T20:39:48Z

Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> Image:LSUkey.png|25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Le..."

New page
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
[[Image:LSUkey.png|25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions]]
</td>
<td align="right">
<math>
\int_a^b\biggl[(\nu - \sigma)(\nu + \sigma + 1) + (\rho^2 - \mu^2)(1 - z^2)^{-1} \biggr] w_\nu^\mu ~w_\sigma^\rho ~dz
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl[
z(\nu-\sigma) w_\nu^\mu ~w_\sigma^\rho + (\sigma+\rho) w_\nu^\mu ~ w_{\sigma-1}^\rho - (\nu + \mu) w_{\nu - 1}^\mu ~w_\sigma^\rho
\biggr]_a^b
</math>
</td>
</tr>
<tr>
<td align="center" colspan=4>
[https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erdélyi (1953)]:  Volume I, §3.12, p. 169, eq. (1)
</td>
</tr>
</table>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">where, <math>w_\nu^\mu(z)</math> and <math>w_\sigma^\rho(z)</math> denote any solutions of Legendre's differential equation</td>
</tr>
</table>