Difference between revisions of "User:Tohline/Appendix/Mathematics/ToroidalConfusion"
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From [http://adsabs.harvard.edu/abs/2000AN....321..363C H. S. Cohl, J. E. Tohline, A. R. P. Rau, & H. M. Srivastiva (2000, Astronomische Nachrichten, 321, no. 5, 363 - 372)] I find: | From equation (34) of [http://adsabs.harvard.edu/abs/2000AN....321..363C H. S. Cohl, J. E. Tohline, A. R. P. Rau, & H. M. Srivastiva (2000, Astronomische Nachrichten, 321, no. 5, 363 - 372)] I find: | ||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
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</table> | </table> | ||
From [http://hcohl.sdf.org/WHIPPLE.html Howard Cohl's online overview] of toroidal functions, I find: | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~Q^n_{m- 1 / 2}(\cosh\alpha)</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~(-1)^n | |||
~\Gamma(n-m + \tfrac{1}{2}) \biggl[ \frac{\pi}{2\sinh\alpha} \biggr]^{1 / 2} P^m_{n- 1 / 2}(\coth\alpha)\, , | |||
</math> | |||
</td> | |||
</tr> | |||
</table> | |||
=See Also= | =See Also= | ||
Revision as of 15:54, 9 May 2018
Confusion Regarding Whipple Formulae
May, 2018 (J.E.Tohline): I am trying to figure out what the correct relationship is between half-integer degree, associated Legendre functions of the first and second kinds. In order to illustrate my current confusion, here I will restrict my presentation to expressions that give <math>~Q^m_{n - 1 / 2}(\cosh\eta)</math> in terms of <math>~P^n_{m - 1 / 2}(\coth\eta)</math>.
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From equation (34) of H. S. Cohl, J. E. Tohline, A. R. P. Rau, & H. M. Srivastiva (2000, Astronomische Nachrichten, 321, no. 5, 363 - 372) I find:
|
<math>~Q^m_{n - 1 / 2}(\cosh\eta)</math> |
<math>~=</math> |
<math>~ \frac{(-1)^n \pi}{\Gamma(n - m + \tfrac{1}{2})} \biggl[ \frac{\pi}{2\sinh\eta} \biggr]^{1 / 2} P^n_{m - 1 / 2}(\coth\eta) \, . </math> |
From Howard Cohl's online overview of toroidal functions, I find:
|
<math>~Q^n_{m- 1 / 2}(\cosh\alpha)</math> |
<math>~=</math> |
<math>~(-1)^n ~\Gamma(n-m + \tfrac{1}{2}) \biggl[ \frac{\pi}{2\sinh\alpha} \biggr]^{1 / 2} P^m_{n- 1 / 2}(\coth\alpha)\, , </math> |
See Also
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