Difference between revisions of "User:Tohline/Appendix/Mathematics/ToroidalConfusion"
(Created page with '<!-- __NOTOC__ will force TOC off --> __FORCETOC__ =Confusion Regarding Whipple Formulae= {{LSU_HBook_header}} =See Also= {{LSU_HBook_footer}}') |
|||
| Line 3: | Line 3: | ||
=Confusion Regarding Whipple Formulae= | =Confusion Regarding Whipple Formulae= | ||
May, 2018: 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>. | |||
{{ | 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: | ||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~Q^m_{n - 1 / 2}(\cosh\eta)</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<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> | |||
</td> | |||
</tr> | |||
</table> | |||
{{LSU_HBook_header}} | |||
=See Also= | =See Also= | ||
Revision as of 15:50, 9 May 2018
Confusion Regarding Whipple Formulae
May, 2018: 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>.
From 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> |
|
|---|
| | Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
See Also
|
|
|---|
|
© 2014 - 2021 by Joel E. Tohline |