Synopsis of Toroidal Coordinate Approach

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>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math>

and,

<math>~\cos\theta</math>

where,

<math>~\equiv</math>

<math>~(\varpi + a)^2 + (z-Z_0)^2 \, ,</math>

and,

<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>~\frac{a \sinh\eta }{(\cosh\eta - \cos\theta)} \, ,</math>

and,

<math>~\frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \, .</math>

The three-dimensional differential volume element is,

<math>\varpi d\varpi ~dz ~d\psi</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,

and,

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,

Also, given that,

<math>~\frac{1}{2}\biggl[ e^\eta + e^{-\eta} \biggr]</math>

and,

<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>

\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>~-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>~(-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>~ 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>~ \dfrac{ \pi/\sqrt{2} }{\left[ \cosh\xi-\cos(\theta - \theta^') \right]^{\frac{1}{2}} } \, . </math>

Next thought …

New Insight

Identical Green's Function Expressions

Caltech's electronic version of A. Erdélyi's (1953) Higher Transcendental Functions; in particular, §3.11, p. 169 of Volume I gives,

<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>

This is valid for,

<math>~t, t^'</math> real

<math>~\nu \ne -1, -2, -3, </math> …

<math>~\psi</math> real.

If we make the association, <math>~t \leftrightarrow \coth\eta</math>, then we also have,

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=0}^\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>~ \sum_{n=0}^\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>~(-1)^n \frac{\pi^{3/2}}{\sqrt{2} \Gamma(n-m+\frac{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+\frac{1}{2})} (\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta) \, . </math>
	<math>~=</math>	<math>~ (-1)^n \sqrt{ \frac{\pi}{2} } ~\Gamma(n - m + \tfrac{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>~\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=0}^\infty \epsilon_n (-1)^n \biggl\{(-1)^n \sqrt{ \frac{\pi}{2} } ~\Gamma(n - m + \tfrac{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{\sinh\eta^'} \sqrt{\sinh\eta} \sum_{n=0}^\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{1}{a\pi} [ (\cosh\eta^' - \cos\theta^') (\cosh\eta - \cos\theta)]^{1 / 2} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] \sum_{n=0}^\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{1}{a\pi} [ (\cosh\eta^' - \cos\theta^') (\cosh\eta - \cos\theta)]^{1 / 2} \sum_{m=0}^{\infty} \sum_{n=0}^\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 match precisely!

Integral Over Polar Angle

Returning to A. Erdélyi's (1953) Higher Transcendental Functions …

Equation (5) in §3.7, p. 155 of Volume I gives,

<math>~ e^{i \mu \pi} ~2^{-\nu - 1} \frac{\Gamma(\nu + \mu + 1) }{\Gamma(\nu + 1) } (z^2 - 1)^{-\mu/2} \int_0^\pi (z+\cos t)^{\mu - \nu - 1} (\sin t)^{2\nu + 1} dt \, . </math>

This is valid for,

<math>~\mathrm{Re} ~\nu > -1</math>

and

<math>~\mathrm{Re} (\nu + \mu + 1) > 0 \, .</math>

Equation (10) in §3.7, p. 156 of Volume I gives,

<math>~ e^{i \mu \pi} ~ (2\pi)^{-\frac{1}{2}} (z^2-1)^{\mu/2} ~\Gamma(\mu + \tfrac{1}{2})~\biggl\{ \int_0^\pi (z - \cos t)^{-\mu - \frac{1}{2}} \cos[(\nu + \tfrac{1}{2})t] ~dt -\cos(\nu\pi) \int_0^\infty (z + \cosh t)^{-\mu - \frac{1}{2}} e^{-(\nu + \frac{1}{2})t} ~dt \biggr\} </math>

This is valid for,

<math>~\mathrm{Re} ~\nu > -\tfrac{1}{2}</math>

and

<math>~\mathrm{Re} (\nu + \mu + 1) > 0 \, .</math>

We will lean on this integral definition of the Legendre function, <math>~Q^\mu_\nu</math>, to evaluate the definite integral in equation (2.56) of Wong (1973), viz.,

<math>~\int_{-\pi}^{\pi} \frac{\cos[n(\theta - \theta^')] d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} </math>

<math>~\int_{-\pi}^{\pi} \frac{ \{\cos(n\theta)\cos(n\theta^') + \sin(n\theta)\sin(n\theta^') \} d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} \, .</math>

Setting, <math>~z = \cosh\eta</math>, <math>~t = \theta</math>, <math>~\mu = 2</math>, and, <math>~\nu = n - \tfrac{1}{2}</math>, we have,

<math>~Q_{n - \frac{1}{2}}^2 (\cosh\eta)</math>	<math>~=</math>	<math>~ (2\pi)^{-\frac{1}{2}} (\cosh^2\eta-1) ~\Gamma(\tfrac{5}{2})~\biggl\{ \int_0^\pi (\cosh\eta - \cos \theta)^{-\frac{5}{2}} \cos(n\theta) ~d\theta -\cos[(n-\tfrac{1}{2})\pi] \int_0^\infty (\cosh\eta + \cosh \theta)^{- \frac{5}{2}} e^{-n\theta} ~d\theta \biggr\} </math>
<math>~\Rightarrow ~~~ \cos(n\theta^') \int_0^\pi \frac{ \cos(n\theta)}{ (\cosh\eta - \cos \theta)^{\frac{5}{2}} }~d\theta </math>	<math>~=</math>	<math>~\frac{ (2\pi)^{\frac{1}{2}} Q_{n - \frac{1}{2}}^2 (\cosh\eta) \cos(n\theta^')}{ (\cosh^2\eta-1) ~\Gamma(\tfrac{5}{2})~ } + \cos(n\theta^')\cos[(n-\tfrac{1}{2})\pi] \int_0^\infty (\cosh\eta + \cosh \theta)^{- \frac{5}{2}} e^{-n\theta} ~d\theta </math>
	<math>~=</math>	<math>~\biggl[ \frac{ 2^2 \sqrt{2} }{ 3 } \biggr] \frac{ Q_{n - \frac{1}{2}}^2(\cosh\eta) \cos(n\theta^')}{ \sinh^2\eta } + \cos(n\theta^')\cos[(n-\tfrac{1}{2})\pi] \int_0^\infty \frac{ d\theta }{ (\cosh\eta + \cosh \theta)^{\frac{5}{2}} e^{n\theta} } \, . </math>

where we have set,

<math>~ \Gamma(\tfrac{5}{2}) = \Gamma(\tfrac{1}{2} + 2) = \frac{ \sqrt{\pi} \cdot 4! }{4^2 \cdot 2!} = \frac{\sqrt{\pi} \cdot 2^3\cdot 3}{ 2^5 } = \frac{3 \sqrt{\pi}}{2^2} \, . </math>

User:Tohline/Appendix/Mathematics/ToroidalSynopsis01

Contents

Synopsis of Toroidal Coordinate Approach

Basics

Arguments of Q and K

Potential

Volume Element

Green's Function

New Insight

Identical Green's Function Expressions

Integral Over Polar Angle

See Also

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