Template:Math/EQ Toroidal03 - Revision history

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link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions $Q_\nu^\mu(z)$ $=$ $e^{i \mu \pi} ~ (2\pi)^{-\frac{1}{2}} (z^2-1)^{\mu/2} ~\Gamma(\mu + \tfrac{1}{2})~\biggl\{ \int_0^\pi (z - \cos t)^{-\mu - \frac{1}{2}} \cos[(\nu +..."$
2024-06-29T23:47:43Z

Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions </td> <td align="right"> <math>Q_\nu^\mu(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <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 +..."

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[[Image:LSUkey.png|25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions]]
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<math>Q_\nu^\mu(z)</math>
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<math>=</math>
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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\}
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[https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erdélyi (1953)]:  Volume I, §3.7, p. 156, eq. (10)
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<td align="right">Valid for:    </td>
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<math>\mathrm{Re} ~\nu > -\tfrac{1}{2}</math> 
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    and    
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<math>\mathrm{Re} (\nu + \mu + 1) > 0 \, .</math>
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