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:
| Expression #1 | ||
|---|---|---|
|
<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:
| Expression #2 | ||
|---|---|---|
|
<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,
| Expression #3 | ||
|---|---|---|
|
<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> |
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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> |
That is, we have,
| Expression #4 | ||
|---|---|---|
|
<math>~Q_{n-1 / 2}^m (\cosh\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 #1 drawn from Cohl et al. (2000), but which appears not to match either of the other two "published" (online) formulae, expressions #2 or #3.
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 #3 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.
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Demonstration that <math>~Q_{-\frac{1}{2}}</math> can be derived from <math>~P_{-\frac{1}{2}}</math> |
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Copying equation (34) from Cohl et al. (2000), we begin with,
then setting <math>~m = n = 0</math>, we have,
Step #1: Associate … <math>z \leftrightarrow \cosh\eta</math>. Then,
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,
Step #3: Again, making the association … <math>z \leftrightarrow \cosh\eta</math>, means,
This, indeed, matches eq. (8.13.4) from Abramowitz & Stegun (1995). |
Cohl's Response to My (May 2018) Email Query
Most of the confusion expressed above stems from the DLMF's use of bold fonts, such as the function on the left-hand side of 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> |
What has been missing in my discussion is an appreciation of the following relationship between bold and plain-text function names,
<math> \boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=e^{-\mu\pi i}\frac{Q^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}. </math>
After making the substitutions, <math>~\mu \rightarrow m</math> and <math>~\nu \rightarrow (n-\tfrac{1}{2})</math>, with this supstitution the Whipple formula becomes,
|
<math>~e^{-m\pi i}\frac{Q^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)}{\Gamma\left(n+m+\tfrac{1}{2}\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> |
|
<math>~\Rightarrow ~~~ Q^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)</math> |
<math>~=</math> |
<math>~e^{m\pi i} \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> |
Still confused!!
|
<math>~Q^\mu_\nu(\cosh\eta)</math> |
<math>~=</math> |
<math>~ \sqrt{\frac{\pi}{2}} ~\Gamma(\nu + \mu + 1) ~e^{i\mu\pi} \biggl[ \frac{1}{\sinh^2\eta} \biggr]^{1 / 4} P^{-\nu-\frac{1}{2}}_{-\mu - \frac{1}{2}} (\coth\eta) </math> |
Making the pair of substitutions,
|
<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> |
we also have,
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<math>~\nu + \mu +1</math> |
<math>~=</math> |
<math>~n - \frac{1}{2} + m + 1</math> |
<math>~=</math> |
<math>~n + m + \frac{1}{2} \, ,</math> |
|
<math>~-\mu - \frac{1}{2}</math> |
<math>~=</math> |
<math>~-m-\frac{1}{2} \, ,</math> |
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<math>~-\nu - \frac{1}{2}</math> |
<math>~=</math> |
<math>~-\biggl(n - \frac{1}{2}\biggr)-\frac{1}{2} </math> |
<math>~=</math> |
<math>~-n \, , </math> |
|
<math>~e^{i\mu\pi}</math> |
<math>~=</math> |
<math>~e^{i m \pi}</math> |
<math>~=</math> |
<math>~(-1)^{m} \, , </math> |
in which case,
|
<math>~Q^m_{n-\frac{1}{2}}(\cosh\eta)</math> |
<math>~=</math> |
<math>~ \sqrt{\frac{\pi}{2}} ~\Gamma(n+m + \tfrac{1}{2}) ~(-1)^m\biggl[ \frac{1}{ \sqrt{\sinh\eta}} \biggr] P^{-n}_{-m - \frac{1}{2}} (\coth\eta) \, . </math> |
Now, since,
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<math>~P^\mu_\nu(z)</math> |
<math>~=</math> |
<math>~P^\mu_{-\nu-1}(z) \, ,</math> |
if we make the substitution,
|
<math>~-(\nu + 1)</math> |
<math>~\rightarrow</math> |
<math>~-(m+\tfrac{1}{2})</math> |
<math>~\Rightarrow</math> |
<math>~\nu</math> |
<math>~\rightarrow</math> |
<math>~m - \tfrac{1}{2} \, ,</math> |
we also know that,
|
<math>~P^\mu_{m-\frac{1}{2}}(z)</math> |
<math>~=</math> |
<math>~P^\mu_{-m-\frac{1}{2}}(z) \, .</math> |
Hence, we can write,
|
<math>~Q^m_{n-\frac{1}{2}}(\cosh\eta)</math> |
<math>~=</math> |
<math>~ \sqrt{\frac{\pi}{2}} ~\Gamma(n+m + \tfrac{1}{2}) ~(-1)^m\biggl[ \frac{1}{ \sqrt{\sinh\eta}} \biggr] P^{-n}_{m - \frac{1}{2}} (\coth\eta) \, . </math> |
Another relation states that, for <math>~n \in \mathbb{N}_0</math>,
|
<math>~P^{-n}_{m-\frac{1}{2}}(z)</math> |
<math>~=</math> |
<math>~\biggl[ \frac{\Gamma(m-n+\frac{1}{2})}{\Gamma(m+n+\frac{1}{2})} \biggr] P^n_{m-\frac{1}{2}}(z) \, .</math> |
So, we obtain,
|
<math>~Q^m_{n-\frac{1}{2}}(\cosh\eta)</math> |
<math>~=</math> |
<math>~(-1)^m \sqrt{\frac{\pi}{2}} ~\Gamma(n+m + \tfrac{1}{2}) \biggl[ \frac{1}{ \sqrt{\sinh\eta}} \biggr] \biggl[ \frac{\Gamma(m-n+\frac{1}{2})}{\Gamma(m+n+\frac{1}{2})} \biggr]P^{n}_{m - \frac{1}{2}} (\coth\eta) \, . </math> |
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|
<math>~=</math> |
<math>~(-1)^m \sqrt{\frac{\pi}{2}} ~\Gamma(m-n+\tfrac{1}{2}) \biggl[ \frac{1}{ \sqrt{\sinh\eta}} \biggr] P^{n}_{m - \frac{1}{2}} (\coth\eta) \, . </math> |
Finally, then, setting <math>~n = m = 0</math>, gives the following sought-for relationship:
|
<math>~Q^0_{-\frac{1}{2}}(\cosh\eta)</math> |
<math>~=</math> |
<math>~ \sqrt{\frac{\pi}{2}} ~\Gamma(\tfrac{1}{2}) \biggl[ \frac{1}{ \sqrt{\sinh\eta}} \biggr] P^{0}_{- \frac{1}{2}} (\coth\eta) \, . </math> |
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<math>~=</math> |
<math>~ \frac{\pi}{\sqrt{2}} ~ \biggl[ \frac{1}{ \sqrt{\sinh\eta}} \biggr] P^{0}_{- \frac{1}{2}} (\coth\eta) \, . </math> |
See Also
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