User:Tohline/Appendix/Mathematics/ToroidalConfusion

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>.

Published Expressions

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>

Copying the Whipple's formula from §14.19 of DLMF,

<math>~\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)</math>

<math>~=</math>

<math>~ \frac{\Gamma\left(m-n+ \tfrac{1}{2}\right)}{\Gamma\left(m+n+\tfrac{1}{2}\right)}\left(\frac{\pi}{2 \sinh\xi}\right)^{1 / 2}P^{n}_{m-\frac{1}{2}}\left(\coth\xi\right) \, . </math>

So far, this gives me three similar but not identical formulae for the same function mapping! As per equation (8) in (yet another source!) A. Gil, J. Segura, & N. M. Temme (2000, JCP, 161, 204 - 217), the relationship is:

<math>~Q_{n-1 / 2}^m (\lambda)</math>

<math>~=</math>

<math>~(-1)^n \frac{\pi^{3/2}}{\sqrt{2} \Gamma(n-m+1 / 2)} (x^2-1)^{1 / 4} P_{m-1 / 2}^n(x) \, , </math>

where, <math>~\lambda \equiv x/\sqrt{x^2-1}</math>. This expression from Gil et al. (2000) means, for example, that by identifying <math>~x</math> with <math>~\coth\eta</math>, we have <math>~\lambda = \cosh\eta</math>, and,

<math>~Q_{n-1 / 2}^m (\cosh\eta)</math>	<math>~=</math>	<math>~(-1)^n \frac{\pi^{3/2}}{\sqrt{2} \Gamma(n-m+1 / 2)} (\coth^2\eta-1)^{1 / 4} P_{m-1 / 2}^n(\coth\eta) </math>
	<math>~=</math>	<math>~ \frac{(-1)^n ~\pi}{\Gamma(n-m+1 / 2)} \biggl( \frac{\pi}{2}\biggr)^{1 / 2} \biggl[\frac{\cosh^2\eta}{\sinh^2\eta}-1 \biggr]^{1 / 4} P_{m-1 / 2}^n(\coth\eta) </math>
	<math>~=</math>	<math>~ \frac{(-1)^n ~\pi}{\Gamma(n-m+1 / 2)} \biggl( \frac{\pi}{2}\biggr)^{1 / 2} \biggl[\frac{1}{\sinh\eta}\biggr]^{1 / 2} P_{m-1 / 2}^n(\coth\eta) </math>
	<math>~=</math>	<math>~ \frac{(-1)^n ~\pi}{\Gamma(n-m+1 / 2)} \biggl[\frac{\pi}{2\sinh\eta}\biggr]^{1 / 2} P_{m-1 / 2}^n(\coth\eta) \, , </math>

which matches the above expression drawn from Cohl et al. (2000), but which does not match either of the other two "published" (online) formulae.

Specific Application

I stumbled into this dilemma when I tried to explicitly demonstrate how <math>~Q_{-1 / 2}(\cosh\eta)</math> can be derived from <math>~P_{-1 / 2}(z)</math> where, from §8.13 of M. Abramowitz & I. A. Stegun (1995), we find,

<math>~Q_{-1 / 2}(\cosh\eta)</math>	<math>~=</math>	<math>~2 e^{- \eta / 2} ~K(e^{-\eta} ) \, , </math>
Abramowitz & Stegun (1995), eq. (8.13.4)

and,

<math>~P_{-1 / 2}(z)</math>	<math>~=</math>	<math>~ \frac{2}{\pi} \biggl[\frac{2}{z+1}\biggr]^{1 / 2} ~K\biggl( \sqrt{ \frac{z-1}{z+1}} \biggr) \, . </math>
Abramowitz & Stegun (1995), eq. (8.13.1)

When I used the Whipple formula as defined in §14.19 of DLMF (expression reprinted above), the function mapping gave me the wrong result; I was off by a factor of <math>~\Gamma(\tfrac{1}{2}) =\sqrt{\pi}</math>. But, as demonstrated below, the Whipple formula provided by Cohl et al. (2000) and by Gil et al. (2000) does give the correct result.

Demonstration that <math>~Q_{-\frac{1}{2}}</math> can be derived from <math>~P_{-\frac{1}{2}}</math>

Copying equation (34) from Cohl et al. (2000), we begin with, <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> then setting <math>~m = n = 0</math>, we have, <math>~Q_{-\frac{1}{2}}(\cosh\eta)</math> <math>~=</math> <math>~ \frac{\pi}{\Gamma(\tfrac{1}{2})} \biggl[ \frac{\pi}{2\sinh\eta} \biggr]^{1 / 2} P_{-\frac{1}{2}}(\coth\eta) </math>   <math>~=</math> <math>~ \frac{\pi}{\sqrt{2}} \biggl[ \frac{1}{\sinh\eta} \biggr]^{1 / 2} P_{-\frac{1}{2}}(\coth\eta) \, . </math> Step #1:   Associate … <math>z \leftrightarrow \cosh\eta</math>. Then, <math>~Q_{-\frac{1}{2}}(\cosh\eta)</math> <math>~=</math> <math>~ \frac{\pi}{\sqrt{2}} \biggl[ \frac{1}{\sqrt{z^2-1}} \biggr]^{1 / 2} P_{-\frac{1}{2}}\biggl(\frac{z}{\sqrt{z^2-1}} \biggr) \, . </math> Step #2:   Now making the association … <math>\Lambda \leftrightarrow z/\sqrt{z^2-1}</math>, and drawing on eq. (8.13.1) from Abramowitz & Stegun (1995), we can write, <math>~P_{-\frac{1}{2}}(\Lambda)</math> <math>~=</math> <math>~ \frac{2}{\pi} \biggl[\frac{2}{\Lambda+1}\biggr]^{1 / 2} ~K\biggl( \sqrt{ \frac{\Lambda-1}{\Lambda+1}} \biggr) </math>   <math>~=</math> <math>~ \frac{2}{\pi} \biggl[\frac{2\sqrt{z^2-1} }{z+\sqrt{z^2-1} }\biggr]^{1 / 2} ~K\biggl( \sqrt{ \frac{z-\sqrt{z^2-1} }{z+\sqrt{z^2-1} }} \biggr) \, . </math> Step #3:   Again, making the association … <math>z \leftrightarrow \cosh\eta</math>, means, <math>~P_{-\frac{1}{2}}(\Lambda)</math> <math>~=</math> <math>~ \frac{2}{\pi} \biggl[\frac{2\sinh\eta }{\cosh\eta +\sinh\eta }\biggr]^{1 / 2} ~K\biggl( \sqrt{ \frac{\cosh\eta-\sinh\eta }{\cosh\eta +\sinh\eta }} \biggr) </math> <math>~\Rightarrow ~~~ Q_{-\frac{1}{2}}(\cosh\eta)</math> <math>~=</math> <math>~ \frac{\pi}{\sqrt{2}} \biggl[ \frac{1}{\sinh\eta} \biggr]^{1 / 2} \frac{2}{\pi} \biggl[\frac{2\sinh\eta }{\cosh\eta +\sinh\eta }\biggr]^{1 / 2} ~K\biggl( \sqrt{ \frac{\cosh\eta-\sinh\eta }{\cosh\eta +\sinh\eta }} \biggr) </math>   <math>~=</math> <math>~ 2 \biggl[\frac{1 }{\cosh\eta +\sinh\eta }\biggr]^{1 / 2} ~K\biggl( \sqrt{ \frac{\cosh^2\eta-\sinh^2\eta }{[\cosh\eta +\sinh\eta ]^2}} ~\biggr) </math>   <math>~=</math> <math>~ 2 \biggl[\frac{1 }{e^\eta }\biggr]^{1 / 2} ~K\biggl( \sqrt{ \frac{1 }{e^{2\eta}}} \biggr) </math>   <math>~=</math> <math>~2 e^{-\eta/2} K(e^{-\eta}) \, . </math> This, indeed, matches eq. (8.13.4) from Abramowitz & Stegun (1995).

Cohl's Response to My Email Query

<math>~Q^\mu_\nu(\cosh\eta)</math>

<math>~=</math>

<math>~ \sqrt{\frac{\pi}{2}} ~\Gamma(\nu + \mu + 1) \biggl[ \frac{1}{\sinh^2\eta} \biggr]^{1 / 4} P^{-\nu-\frac{1}{2}}_{-\mu - \frac{1}{2}} (\coth\eta) </math>

Now substitute:

<math>~\nu</math>	<math>~=</math>	<math>~n - \frac{1}{2} \, ,</math>		<math>~n ~~\in</math>	<math>~\mathbb{N}_0 = \{ 0, 1, 2, \cdots\}</math>
<math>~\mu</math>	<math>~=</math>	<math>~m \, ,</math>		<math>~m ~~\in</math>	<math>~\mathbb{N}_0 = \{ 0, 1, 2, \cdots\}</math>

See Also

© 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