Difference between revisions of "User:Tohline/Appendix/Mathematics/ToroidalSynopsis01"
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=See Also= | =See Also= | ||
* [https://authors.library.caltech.edu/43491/1/Volume%201.pdf Arthur Erdélyi (1953, New York: McGraw-Hill)] — ''Higher Transcendental Functions'', especially Volume I, Chapter 3 (pp. 120 - 174) | |||
* [http://adsabs.harvard.edu/abs/1940QJMat..11..222C T. G. Cowling (1940, Quart. J. Math. Oxford Ser., 11, 222 - 224)] — ''On Certain Expansions Involving Products of Legendre Functions'' | |||
{{LSU_HBook_footer}} | {{LSU_HBook_footer}} |
Revision as of 03:09, 11 June 2018
Synopsis of Toroidal Coordinate Approach
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Basics
Here we attempt to bring together — in as succinct a manner as possible — our approach and C.-Y. Wong's (1973) approach to determining the gravitational potential of an axisymmetric, uniform-density torus that has a major radius, <math>~R</math>, and a minor, cross-sectional radius, <math>~d</math>. The relevant toroidal coordinate system is one based on an anchor ring of major radius,
<math>~a^2 \equiv R^2 - d^2 \, .</math>
If the meridional-plane location of the anchor ring — as written in cylindrical coordinates — is, <math>~(\varpi, z) = (a,Z_0)</math>, then the preferred toroidal-coordinate system has meridional-plane coordinates, <math>~(\eta, \theta)</math>, defined such that,
<math>~\eta</math> |
<math>~=</math> |
<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math> |
and, |
<math>~\cos\theta</math> |
<math>~=</math> |
<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,</math> |
where,
<math>~r_1^2 </math> |
<math>~\equiv</math> |
<math>~(\varpi + a)^2 + (z-Z_0)^2 \, ,</math> |
and, |
<math>~r_2^2 </math> |
<math>~\equiv</math> |
<math>~(\varpi - a)^2 + (z-Z_0)^2 \, ,</math> |
and <math>~\theta</math> has the same sign as <math>~(z-Z_0)</math>. Mapping the other direction, we have,
<math>~\varpi</math> |
<math>~=</math> |
<math>~\frac{a \sinh\eta }{(\cosh\eta - \cos\theta)} \, ,</math> |
and, |
<math>~z-Z_0</math> |
<math>~=</math> |
<math>~\frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \, .</math> |
The three-dimensional differential volume element is,
<math>~d^3 r</math> |
<math>~=</math> |
<math>\varpi d\varpi ~dz ~d\psi</math> |
<math>~=</math> |
<math>~\biggl[ \frac{a^3\sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi \, .</math> |
Note that, if <math>~\eta_0</math> identifies the surface of the uniform-density torus, then,
<math>~\cosh\eta_0</math> |
<math>~=</math> |
<math>~\frac{R}{d} \, ,</math> |
<math>~\sinh\eta_0</math> |
<math>~=</math> |
<math>~\frac{a}{d} \, ,</math> |
and, |
<math>~\coth\eta_0</math> |
<math>~=</math> |
<math>~\frac{R}{a} \, ;</math> |
and when the integral over the volume element is completed — that is, over all <math>~\psi</math>, over all <math>~\theta</math>, and over the "radial" interval, <math>~\eta_0 \le \eta \le \infty</math> — the resulting volume is,
<math>~V</math> |
<math>~=</math> |
<math>~\frac{2\pi^2 \cosh\eta_0}{\sinh^3\eta_0}</math> |
<math>~=</math> |
<math>~2\pi^2 Rd^2 \, .</math> |
Also, given that,
<math>~\cosh\eta</math> |
<math>~=</math> |
<math>~\frac{1}{2}\biggl[ e^\eta + e^{-\eta} \biggr]</math> |
and, |
<math>~\sinh\eta</math> |
<math>~=</math> |
<math>~\frac{1}{2}\biggl[ e^\eta - e^{-\eta} \biggr] \, ,</math> |
we have,
<math>~\coth\eta</math> |
<math>~=</math> |
<math>~\biggl[ e^\eta + e^{-\eta} \biggr]\biggl[ e^\eta - e^{-\eta} \biggr]^{-1}</math> |
<math>~=</math> |
<math>~\biggl[ \frac{r_1}{r_2} + \frac{r_2}{r_1} \biggr]\biggl[ \frac{r_1}{r_2} - \frac{r_2}{r_1} \biggr]^{-1}</math> |
|
<math>~=</math> |
<math>~\biggl[ \frac{r_1^2 + r_2^2}{r_1 r_2} \biggr]\biggl[ \frac{r_1^2 - r_2^2}{r_1 r_2} \biggr]^{-1}</math> |
<math>~=</math> |
<math>~\biggl[ \frac{r_1^2 + r_2^2}{r_1^2 - r_2^2} \biggr]</math> |
|
<math>~=</math> |
<math>~ \frac{ \varpi^2 + a^2 + (z - Z_0)^2 }{ 2a\varpi } \, . </math> |
Arguments of Q and K
Want to explore argument of <math>~Q_{-1 / 2}(\Chi)</math>, namely,
<math> \Chi \equiv \frac{(\varpi^')^2 + \varpi^2 + (z^' - z)^2}{2\varpi^' \varpi} . </math>
Therefore,
<math>~2\varpi \biggl[ \varpi^' \Chi - a\coth\eta\biggr]</math> |
<math>~=</math> |
<math>~ (\varpi^')^2 + \varpi^2 + (z^' - z)^2 - [\varpi^2 + a^2 + (z - Z_0)^2 ] </math> |
|
<math>~=</math> |
<math>~ (\varpi^')^2 - a^2 + [ (z^')^2 - 2z^' z + z^2]- [z^2 - 2zZ_0 + Z_0^2] </math> |
|
<math>~=</math> |
<math>~ (\varpi^')^2 - a^2 + (z^')^2- Z_0^2 +2z(Z_0 - z^' ) </math> |
<math>~\Rightarrow ~~~2a\biggl[ \frac{\sinh\eta }{(\cosh\eta - \cos\theta)} \biggr]\biggl[ \varpi^' \Chi - a\coth\eta\biggr]</math> |
<math>~=</math> |
<math>~ (\varpi^')^2 - a^2 + (z^')^2- Z_0^2 +2(Z_0 - z^' )\biggl[ Z_0 + \frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \biggr] </math> |
|
<math>~=</math> |
<math>~ 2aC_0 +2a(Z_0 - z^' )\biggl[ \frac{\sin\theta}{(\cosh\eta - \cos\theta)} \biggr] </math> |
<math>~\Rightarrow ~~~ \sinh\eta \biggl[ \varpi^' \Chi - a\coth\eta\biggr]</math> |
<math>~=</math> |
<math>~ C_0 (\cosh\eta - \cos\theta) + (Z_0 - z^' ) \sin\theta </math> |
<math>~\Rightarrow ~~~ \varpi^' \Chi </math> |
<math>~=</math> |
<math>~ \frac{1}{\sinh\eta} \biggl[ C_0 (\cosh\eta - \cos\theta) + (Z_0 - z^' ) \sin\theta + a\cosh\eta\biggr] </math> |
<math>~\Rightarrow ~~~ \Chi </math> |
<math>~=</math> |
<math>~ \frac{1}{\varpi^' \sinh\eta} \biggl[ (C_0 + a)\cosh\eta + (Z_0 - z^' ) \sin\theta - C_0 \cos\theta \biggr] </math> |
where,
<math>~ C_0 \equiv \frac{1}{2a}\biggl[ (\varpi^')^2 - a^2 + (z^')^2- Z_0^2 +2Z_0 (Z_0 - z^' ) \biggr] = \frac{1}{2a}\biggl[ (\varpi^')^2 - a^2 + (z^')^2 +Z_0^2 - 2Z_0 z^' \biggr] = \frac{1}{2a}\biggl[ (\varpi^')^2 - a^2 + (z^' - Z_0)^2 \biggr] \, . </math>
Now, notice that,
<math>~ ( \varpi^')^2 + a^2 + (z^' - Z_0)^2 </math> |
<math>~=</math> |
<math>~ 2a\varpi^'~\coth\eta^' </math> |
||
<math>~\Rightarrow ~~~ ( \varpi^')^2 - a^2 + (z^' - Z_0)^2 </math> |
<math>~=</math> |
<math>~ 2a\varpi^'~\coth\eta^' - 2a^2 </math> |
||
<math>~\Rightarrow ~~~ C_0 </math> |
<math>~=</math> |
<math>~ \varpi^'~\coth\eta^' - a </math> |
||
|
<math>~=</math> |
<math>~ \biggl[ \frac{a \sinh\eta^' }{(\cosh\eta^' - \cos\theta^')} \biggr] ~\coth\eta^' - a </math> |
||
|
<math>~=</math> |
<math>~ \biggl[ \frac{a \cosh\eta^' }{(\cosh\eta^' - \cos\theta^')} \biggr] - a \, . </math> |
Hence,
<math>~ \Chi </math> |
<math>~=</math> |
<math>~ \frac{\cosh\eta}{\varpi^' \sinh\eta} \biggl[ \varpi^' \coth\eta^' \biggr] + \frac{1}{\sinh\eta} \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{a \sinh\eta^' } \biggr] \biggl[ (Z_0 - z^' ) \sin\theta - C_0 \cos\theta \biggr] </math> |
|
<math>~=</math> |
<math>~ \coth\eta \cdot \coth\eta^' + \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{a \sinh\eta \cdot \sinh\eta^' } \biggr] \biggl[ (Z_0 - z^' ) \sin\theta - C_0 \cos\theta \biggr] </math> |
|
<math>~=</math> |
<math>~ \coth\eta \cdot \coth\eta^' - \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{a \sinh\eta \cdot \sinh\eta^' } \biggr] \biggl\{ \biggl[ \frac{a \sin\theta^'}{(\cosh\eta^' - \cos\theta^')} \biggr] \sin\theta + \biggl[ \frac{a \cosh\eta^' }{(\cosh\eta^' - \cos\theta^')} \biggr] \cos\theta - a\cos\theta\biggr\} </math> |
|
<math>~=</math> |
<math>~ \coth\eta \cdot \coth\eta^' - \biggl[ \frac{1 }{ \sinh\eta \cdot \sinh\eta^' } \biggr] \biggl\{ \sin\theta^' \sin\theta + \cosh\eta^' \cos\theta - (\cosh\eta^' - \cos\theta^')\cos\theta\biggr\} </math> |
|
<math>~=</math> |
<math>~ \coth\eta \cdot \coth\eta^' - \biggl[ \frac{\sin\theta^' \sin\theta +\cos\theta^'\cos\theta }{ \sinh\eta \cdot \sinh\eta^' } \biggr] </math> |
|
<math>~=</math> |
<math>~ \biggl[ \frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } \biggr] \, . </math> |
Also,
<math>~ \Chi +1 </math> |
<math>~=</math> |
<math>~ \biggl[ \frac{\sinh\eta \cdot \sinh\eta^' + \cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } \biggr] </math> |
|
<math>~=</math> |
<math>~ \biggl[ \frac{ \cosh(\eta^' + \eta) - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } \biggr] </math> |
<math>~ \Rightarrow ~~~\mu^2 \equiv \frac{ 2 }{\Chi +1 }</math> |
<math>~=</math> |
<math>~ \biggl[ \frac{2 \sinh\eta \cdot \sinh\eta^' }{ \cosh(\eta^' + \eta) - \cos(\theta^' - \theta) } \biggr] \, . </math> |
NOTE by Tohline: On 5 June 2018, I used Excel to test the validity of the toroidal-coordinate-based expressions that have been derived here, and summarized in the following table.
Summary Table |
||
---|---|---|
Quantity |
Raw Expression in Cylindrical Coordinates |
Expression in Terms of Toroidal Coordinates |
<math>~\Chi</math> |
<math> \frac{(\varpi^')^2 + \varpi^2 + (z^' - z)^2}{2\varpi^' \varpi} </math> |
<math>~ \frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } </math> |
<math>~\mu^2 \equiv \frac{2}{\Chi + 1}</math> |
<math> \frac{4\varpi^' \varpi}{(\varpi^' + \varpi)^2 + (z^' - z)^2} </math> |
<math>~ \frac{2 \sinh\eta \cdot \sinh\eta^' }{ \cosh(\eta^' + \eta) - \cos(\theta^' - \theta) } </math> |
Potential
The potential, <math>~U({\vec{r}}~')</math>, at a point <math>~{\vec{r}}~'</math> due to an arbitrary mass distribution, <math>~\rho({\vec{r}})</math>, is,
<math>~U({\vec{r}}~')</math> |
<math>~=</math> |
<math>~-G \iiint \frac{\rho(\vec{r}) d^3r}{|~\vec{r} - {\vec{r}}^{~'} ~|} \, .</math> |
Volume Element
See above.
Green's Function
Wong (1973) points out that in toroidal coordinates the Green's function is,
<math>~\frac{1}{|~\vec{r} - {\vec{r}}^{~'} ~|} </math> |
<math>~=</math> |
<math>~ \frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 } \sum\limits_{m,n} (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} </math> |
|
|
<math>~ \times \cos[m(\psi - \psi^')][\cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta \end{cases}\, , </math> |
Wong (1973), Eq. (2.53) |
where, <math>~P^m_{n-1 / 2}, Q^m_{n-1 / 2}</math> are "Legendre functions of the first and second kind with order <math>~n - \tfrac{1}{2}</math> and degree <math>~m</math> (toroidal harmonics)," and <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0 = 1</math> and <math>~\epsilon_m = 2</math> for all <math>~m \ge 1</math>. According to CT99, the Green's function written in toroidal coordinates is,
<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}</math> |
<math>~=</math> |
<math>~ \frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) </math> |
|
<math>~=</math> |
<math>~ \frac{1}{a\pi} \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{\sinh\eta^' } \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) \, . </math> |
Things to note:
- The argument of <math>~Q_{m - 1 / 2}</math> in the CT99 expression is very different from the argument of <math>~Q^m_{n - 1 / 2}</math> (or <math>~P^m_{n - 1 / 2}</math>) in Wong's expression.
- In both expressions, <math>~m</math> is the integer multiplying the azimuthal angle, <math>~\psi</math>, but in the CT99 expression this index serves as the subscript index of the function, <math>~Q</math>, whereas in Wong's expression it serves as the superscript index of both functions, <math>~Q</math> and <math>~P</math>. In this context, note that,
<math>~Q^m_{n-\frac{1}{2}}(\cosh\eta)</math>
<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>
- Wong's expression contains not only a summation over the index, <math>~m</math>, but also an explicit summation over the index, <math>~n</math>, which multiplies the "polar" angle, <math>~\theta</math>; no such additional summation appears in the CT99 expression, indicating that the summation over <math>~n</math> has implicitly already been completed. In this context, note that the summation expression gives,
<math>~ Q^{\mu}_{-\frac{1}{2}}\left(\cosh\xi\right) + 2\sum_{n=1}^{\infty} Q^{\mu}_{n-\frac{1}{2}}\left(\cosh\xi\right) \cos\left[ n (\theta - \theta^') \right] </math>
<math>~=</math>
<math>~ e^{\mu\pi i} \Gamma\left(\mu+ \tfrac{1}{2} \right) \biggl[ \dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\sinh\xi\right)^{\mu }}{\left\{ \cosh\xi -\cos\left[ n (\theta - \theta^') \right] \right\}^{\mu+(1/2)}}\biggr] \, ; </math>
or, specifically for the case of <math>~\mu = 0</math>,
<math>~ \sum_{n=0}^{\infty} \epsilon_n Q_{n-\frac{1}{2}}\left(\cosh\xi\right) \cos\left[ n(\theta - \theta^') \right] </math>
<math>~=</math>
<math>~ \dfrac{ \pi/\sqrt{2} }{\left[ \cosh\xi-\cos(\theta - \theta^') \right]^{\frac{1}{2}} } \, . </math>
- Next thought …
New Insight
Caltech's electronic version §3.11, p. 169 of Volume I gives,
<math>~ Q_\nu[t t^' - (t^2-1)^{1 / 2} (t^{'2} - 1)^{1 / 2} \cos\psi] </math> |
<math>~=</math> |
<math>~ Q_\nu(t) P_\nu(t^') + 2\sum_{n=1}^\infty (-1)^n Q^n_\nu(t) P^{-n}_\nu(t^') \cos(n\psi) \, . </math> |
If we make the association, <math>~t \leftrightarrow \coth\eta</math>, then we also have,
<math>~\frac{1}{\sinh\eta}</math> |
<math>~=</math> |
<math>~\sqrt{t^2 - 1} \, ,</math> |
in which case,
<math>~ \Chi </math> |
<math>~=</math> |
<math>~ \frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } </math> |
|
<math>~=</math> |
<math>~ t t^' - (t^2-1)^{1 / 2}(t^{'2}-1)^{1 / 2}\cos(\theta^' - \theta) \, . </math> |
Put together, then, these expressions mean,
<math>~ Q_{m - 1 / 2}(\Chi) </math> |
<math>~=</math> |
<math>~ Q_{m-1 / 2}(\coth\eta) P_{m - 1 / 2}(\coth\eta^') + 2\sum_{n=1}^\infty (-1)^n Q^n_{m - 1 / 2}(\coth\eta) P^{-n}_{m - 1 / 2}(\coth\eta^') \cos[n(\theta^' - \theta)] </math> |
|
<math>~=</math> |
<math>~ \sum_{n=1}^\infty \epsilon_n (-1)^n Q^n_{m - 1 / 2}(\coth\eta) P^{-n}_{m - 1 / 2}(\coth\eta^') \cos[n(\theta^' - \theta)] \, . </math> |
Also, from our derived <math>~Q-P</math> relation,
<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> |
<math>~\Rightarrow ~~~ P^{-n}_{m - \frac{1}{2}} (\coth\eta)</math> |
<math>~=</math> |
<math>~ \sqrt{\frac{2}{\pi}} ~\frac{(-1)^m \sqrt{\sinh\eta} }{\Gamma(n+m + \tfrac{1}{2})} ~ Q^m_{n-\frac{1}{2}}(\cosh\eta) \, . </math> |
we can write,
<math>~ Q_{m - 1 / 2}(\Chi) </math> |
<math>~=</math> |
<math>~ \sum_{n=1}^\infty \epsilon_n (-1)^n Q^n_{m - 1 / 2}(\coth\eta) \biggl\{ \sqrt{\frac{2}{\pi}} ~\frac{(-1)^m \sqrt{\sinh\eta^'} }{\Gamma(n+m + \tfrac{1}{2})} ~ Q^m_{n-\frac{1}{2}}(\cosh\eta^') \biggr\} \cos[n(\theta^' - \theta)] </math> |
Next, we pull from the accompanying discussion of the Gil et al. (2000) expression,
<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>. Identifying <math>~x</math> with <math>~\cosh\eta</math>, in which case we have <math>~\lambda = \coth\eta</math>, and, switching index notation, <math>~n \leftrightarrow m</math>, gives,
<math>~Q_{m-1 / 2}^n (\coth\eta)</math> |
<math>~=</math> |
<math>~(-1)^m \frac{\pi^{3/2}}{\sqrt{2} \Gamma(m-n+1 / 2)} (\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta) \, . </math> |
|
<math>~=</math> |
<math>~ (-1)^n \pi^{1 / 2} \Gamma(n - m + 1 / 2)(\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta) \, . </math> |
where, this last step also incorporates the "Euler reflection formula for gamma functions", namely,
<math>~\frac{1}{\Gamma(m-n+\tfrac{1}{2})} </math> |
<math>~=</math> |
<math>~\frac{\Gamma(n-m+\frac{1}{2}) }{\pi (-1)^{m+n}} \, .</math> |
So we have,
<math>~ Q_{m - 1 / 2}(\Chi) </math> |
<math>~=</math> |
<math>~ \sum_{n=1}^\infty \epsilon_n (-1)^n \biggl\{ (-1)^n \pi^{1 / 2} \Gamma(n - m + 1 / 2)(\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta) \biggr\} \biggl\{ \sqrt{\frac{2}{\pi}} ~\frac{(-1)^m \sqrt{\sinh\eta^'} }{\Gamma(n+m + \tfrac{1}{2})} ~ Q^m_{n-\frac{1}{2}}(\cosh\eta^') \biggr\} \cos[n(\theta^' - \theta)] </math> |
|
<math>~=</math> |
<math>~\sqrt{2}~\sqrt{\sinh\eta^'} \sqrt{\sinh\eta} \sum_{n=1}^\infty \epsilon_n (-1)^m \frac{ \Gamma(n - m + \tfrac{1}{2})}{\Gamma(n+m + \tfrac{1}{2})} P_{n-1 / 2}^m(\cosh\eta) Q^m_{n-\frac{1}{2}}(\cosh\eta^') \cos[n(\theta^' - \theta)] \, . </math> |
Hence, the CT99 Green's function may be rewritten as,
<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}</math> |
<math>~=</math> |
<math>~ \frac{\sqrt{2}}{a\pi} [ (\cosh\eta^' - \cos\theta^') (\cosh\eta - \cos\theta)]^{1 / 2} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] \sum_{n=1}^\infty \epsilon_n (-1)^m \frac{ \Gamma(n - m + \tfrac{1}{2})}{\Gamma(n+m + \tfrac{1}{2})} P_{n-1 / 2}^m(\cosh\eta) Q^m_{n-\frac{1}{2}}(\cosh\eta^') \cos[n(\theta^' - \theta)] </math> |
|
<math>~=</math> |
<math>~ \frac{\sqrt{2}}{a\pi} [ (\cosh\eta^' - \cos\theta^') (\cosh\eta - \cos\theta)]^{1 / 2} \sum_{m=0}^{\infty} \sum_{n=1}^\infty \epsilon_m\epsilon_n (-1)^m \frac{ \Gamma(n - m + \tfrac{1}{2})}{\Gamma(n+m + \tfrac{1}{2})} \cos[m(\psi - \psi^')] \cos[n(\theta^' - \theta)] P_{n-1 / 2}^m(\cosh\eta) Q^m_{n-\frac{1}{2}}(\cosh\eta^') \, . </math> |
Let's compare this with Wong's (1973) Green's function, namely,
<math>~\frac{1}{|~\vec{r} - {\vec{r}}^{~'} ~|} </math> |
<math>~=</math> |
<math>~ \frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 } \sum\limits_{m,n} (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} </math> |
|
|
<math>~ \times \cos[m(\psi - \psi^')][\cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta \end{cases}\, , </math> |
Wong (1973), Eq. (2.53) |
[June 10, 2018] Amazing! The two expressions differ only by the leading factor of <math>~\sqrt{2}</math>
See Also
- Arthur Erdélyi (1953, New York: McGraw-Hill) — Higher Transcendental Functions, especially Volume I, Chapter 3 (pp. 120 - 174)
- T. G. Cowling (1940, Quart. J. Math. Oxford Ser., 11, 222 - 224) — On Certain Expansions Involving Products of Legendre Functions
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