Template:Math/EQ Toroidal08 - Revision history

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link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions $Q_{-\frac{1}{2}}^\mu(z) + 2\sum_{n=1}^\infty Q^\mu_{n-\frac{1}{2}}(z) \cos(n\nu)$ $=$ $e^{i\mu\pi}~\biggl(\frac{\pi}{2} \biggr)^{1 / 2} \Gamma(\mu + \tfrac{1}{2})\b..."$
2024-06-29T23:51:45Z

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_{-\frac{1}{2}}^\mu(z) + 2\sum_{n=1}^\infty Q^\mu_{n-\frac{1}{2}}(z) \cos(n\nu) </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> e^{i\mu\pi}~\biggl(\frac{\pi}{2} \biggr)^{1 / 2} \Gamma(\mu + \tfrac{1}{2})\b..."

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[[Image:LSUkey.png|25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions]]
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Q_{-\frac{1}{2}}^\mu(z) + 2\sum_{n=1}^\infty Q^\mu_{n-\frac{1}{2}}(z) \cos(n\nu)
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<math>=</math>
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e^{i\mu\pi}~\biggl(\frac{\pi}{2} \biggr)^{1 / 2} \Gamma(\mu + \tfrac{1}{2})\biggl[ \frac{(z^2-1)^{\mu/2}}{(z - \cos\nu)^{\mu + \frac{1}{2}}} \biggr]
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[https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erdélyi (1953)]:  Volume I, §3.10, p. 166, eq. (3)
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<td align="right">Valid for:    </td>
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<math>\mathrm{Re}~\mu > - \tfrac{1}{2}</math>
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