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>.
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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:
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<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:
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<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,
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<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> |
As per equation (8) in A. Gil, J. Segura, & N. M. Temme (2000, JCP, 161, 204 - 217), the relationship is:
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<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,
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<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> |
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<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> |
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<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> |
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<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) expressions.
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,
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<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,
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<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) | ||
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Proof that these are the same expressions: |
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Copying the Whipple's formula from §14.19 of DLMF,
then setting <math>~m = n = 0</math>, we have,
Step #1: Associate … <math>z \leftrightarrow \cosh\xi</math>. Then,
Step #2: Now making the association … <math>\Lambda \leftrightarrow z/\sqrt{z^2-1}</math>, we can write,
Step #3: Again, making the association … <math>z \leftrightarrow \cosh\xi</math>, means,
which, apart from the leading factor of <math>~\pi^{-1 / 2}</math>, exactly matches the above expression. Note: From Howard Cohl's online overview — see, also, below — we find that the Whipple formula is slightly different from the one (quoted above) drawn from DLMF. According to Cohl the Whipple formula should be,
The DLMF expression needs to be multiplied by <math>~(-1)^m\Gamma (m + n + \tfrac{1}{2} )</math> in order to match the expression provided by Cohl; for the case being considered here of <math>~m=n=0</math>, this factor is precisely <math>~\Gamma(\tfrac{1}{2}) = \sqrt{\pi}</math> — see, for example, Wikipedia's discussion of the gamma function — which cancels this confusing factor of <math>~\pi^{-1 / 2}</math>. |
See Also
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