Difference between revisions of "User:Tohline/Appendix/Mathematics/ToroidalConfusion"

From VistrailsWiki
Jump to navigation Jump to search
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>.
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>.
 
 
{{LSU_HBook_header}}
 


From [http://adsabs.harvard.edu/abs/2000AN....321..363C H. S. Cohl, J. E. Tohline, A. R. P. Rau, &amp; H. M. Srivastiva (2000, Astronomische Nachrichten, 321, no. 5, 363 - 372)] I find:
From [http://adsabs.harvard.edu/abs/2000AN....321..363C H. S. Cohl, J. E. Tohline, A. R. P. Rau, &amp; H. M. Srivastiva (2000, Astronomische Nachrichten, 321, no. 5, 363 - 372)] I find:
Line 23: Line 27:
</table>
</table>


{{LSU_HBook_header}}


=See Also=
=See Also=

Revision as of 15:51, 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>.


Whitworth's (1981) Isothermal Free-Energy Surface
|   Tiled Menu   |   Tables of Content   |  Banner Video   |  Tohline Home Page   |


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>


See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
|   H_Book Home   |   YouTube   |
Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS |
Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation