# 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, $~R$, and a minor, cross-sectional radius, $~d$. The relevant toroidal coordinate system is one based on an anchor ring of major radius,

$~a^2 \equiv R^2 - d^2 \, .$

If the meridional-plane location of the anchor ring — as written in cylindrical coordinates — is, $~(\varpi, z) = (a,Z_0)$, then the preferred toroidal-coordinate system has meridional-plane coordinates, $~(\eta, \theta)$, defined such that,

 $~\eta$ $~=$ $~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,$ and, $~\cos\theta$ $~=$ $~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,$

where,

 $~r_1^2$ $~\equiv$ $~(\varpi + a)^2 + (z-Z_0)^2 \, ,$ and, $~r_2^2$ $~\equiv$ $~(\varpi - a)^2 + (z-Z_0)^2 \, ,$

and $~\theta$ has the same sign as $~(z-Z_0)$. Mapping the other direction, we have,

 $~\varpi$ $~=$ $~\frac{a \sinh\eta }{(\cosh\eta - \cos\theta)} \, ,$ and, $~z-Z_0$ $~=$ $~\frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \, .$

The three-dimensional differential volume element is,

 $~d^3 r$ $~=$ $\varpi d\varpi ~dz ~d\psi$ $~=$ $~\biggl[ \frac{a^3\sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi \, .$

Note that, if $~\eta_0$ identifies the surface of the uniform-density torus, then,

 $~\cosh\eta_0$ $~=$ $~\frac{R}{d} \, ,$ $~\sinh\eta_0$ $~=$ $~\frac{a}{d} \, ,$ and, $~\coth\eta_0$ $~=$ $~\frac{R}{a} \, ;$

and when the integral over the volume element is completed — that is, over all $~\psi$, over all $~\theta$, and over the "radial" interval, $~\eta_0 \le \eta \le \infty$ — the resulting volume is,

 $~V$ $~=$ $~\frac{2\pi^2 \cosh\eta_0}{\sinh^3\eta_0}$ $~=$ $~2\pi^2 Rd^2 \, .$

Also, given that,

 $~\cosh\eta$ $~=$ $~\frac{1}{2}\biggl[ e^\eta + e^{-\eta} \biggr]$ and, $~\sinh\eta$ $~=$ $~\frac{1}{2}\biggl[ e^\eta - e^{-\eta} \biggr] \, ,$

we have,

 $~\coth\eta$ $~=$ $~\biggl[ e^\eta + e^{-\eta} \biggr]\biggl[ e^\eta - e^{-\eta} \biggr]^{-1}$ $~=$ $~\biggl[ \frac{r_1}{r_2} + \frac{r_2}{r_1} \biggr]\biggl[ \frac{r_1}{r_2} - \frac{r_2}{r_1} \biggr]^{-1}$ $~=$ $~\biggl[ \frac{r_1^2 + r_2^2}{r_1 r_2} \biggr]\biggl[ \frac{r_1^2 - r_2^2}{r_1 r_2} \biggr]^{-1}$ $~=$ $~\biggl[ \frac{r_1^2 + r_2^2}{r_1^2 - r_2^2} \biggr]$ $~=$ $~ \frac{ \varpi^2 + a^2 + (z - Z_0)^2 }{ 2a\varpi } \, .$

## Arguments of Q and K

Want to explore argument of $~Q_{-1 / 2}(\Chi)$, namely,

$\Chi \equiv \frac{(\varpi^')^2 + \varpi^2 + (z^' - z)^2}{2\varpi^' \varpi} .$

Therefore,

 $~2\varpi \biggl[ \varpi^' \Chi - a\coth\eta\biggr]$ $~=$ $~ (\varpi^')^2 + \varpi^2 + (z^' - z)^2 - [\varpi^2 + a^2 + (z - Z_0)^2 ]$ $~=$ $~ (\varpi^')^2 - a^2 + [ (z^')^2 - 2z^' z + z^2]- [z^2 - 2zZ_0 + Z_0^2]$ $~=$ $~ (\varpi^')^2 - a^2 + (z^')^2- Z_0^2 +2z(Z_0 - z^' )$ $~\Rightarrow ~~~2a\biggl[ \frac{\sinh\eta }{(\cosh\eta - \cos\theta)} \biggr]\biggl[ \varpi^' \Chi - a\coth\eta\biggr]$ $~=$ $~ (\varpi^')^2 - a^2 + (z^')^2- Z_0^2 +2(Z_0 - z^' )\biggl[ Z_0 + \frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \biggr]$ $~=$ $~ 2aC_0 +2a(Z_0 - z^' )\biggl[ \frac{\sin\theta}{(\cosh\eta - \cos\theta)} \biggr]$ $~\Rightarrow ~~~ \sinh\eta \biggl[ \varpi^' \Chi - a\coth\eta\biggr]$ $~=$ $~ C_0 (\cosh\eta - \cos\theta) + (Z_0 - z^' ) \sin\theta$ $~\Rightarrow ~~~ \varpi^' \Chi$ $~=$ $~ \frac{1}{\sinh\eta} \biggl[ C_0 (\cosh\eta - \cos\theta) + (Z_0 - z^' ) \sin\theta + a\cosh\eta\biggr]$ $~\Rightarrow ~~~ \Chi$ $~=$ $~ \frac{1}{\varpi^' \sinh\eta} \biggl[ (C_0 + a)\cosh\eta + (Z_0 - z^' ) \sin\theta - C_0 \cos\theta \biggr]$

where,

$~ C_0 \equiv \frac{1}{2a}\biggl[ (\varpi^')^2 - a^2 + (z^')^2- Z_0^2 +2Z_0 (Z_0 - z^' ) \biggr] = \frac{1}{2a}\biggl[ (\varpi^')^2 - a^2 + (z^')^2 +Z_0^2 - 2Z_0 z^' \biggr] = \frac{1}{2a}\biggl[ (\varpi^')^2 - a^2 + (z^' - Z_0)^2 \biggr] \, .$

Now, notice that,

 $~ ( \varpi^')^2 + a^2 + (z^' - Z_0)^2$ $~=$ $~ 2a\varpi^'~\coth\eta^'$ $~\Rightarrow ~~~ ( \varpi^')^2 - a^2 + (z^' - Z_0)^2$ $~=$ $~ 2a\varpi^'~\coth\eta^' - 2a^2$ $~\Rightarrow ~~~ C_0$ $~=$ $~ \varpi^'~\coth\eta^' - a$ $~=$ $~ \biggl[ \frac{a \sinh\eta^' }{(\cosh\eta^' - \cos\theta^')} \biggr] ~\coth\eta^' - a$ $~=$ $~ \biggl[ \frac{a \cosh\eta^' }{(\cosh\eta^' - \cos\theta^')} \biggr] - a \, .$

Hence,

 $~ \Chi$ $~=$ $~ \frac{\cosh\eta}{\varpi^' \sinh\eta} \biggl[ \varpi^' \coth\eta^' \biggr] + \frac{1}{\sinh\eta} \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{a \sinh\eta^' } \biggr] \biggl[ (Z_0 - z^' ) \sin\theta - C_0 \cos\theta \biggr]$ $~=$ $~ \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] $ $~=$ $~ \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\} $ $~=$ $~ \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\} $ $~=$ $~ \coth\eta \cdot \coth\eta^' - \biggl[ \frac{\sin\theta^' \sin\theta +\cos\theta^'\cos\theta }{ \sinh\eta \cdot \sinh\eta^' } \biggr] $ $~=$ $~ \biggl[ \frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } \biggr] \, .$

Also,

 $~ \Chi +1$ $~=$ $~ \biggl[ \frac{\sinh\eta \cdot \sinh\eta^' + \cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } \biggr]$ $~=$ $~ \biggl[ \frac{ \cosh(\eta^' + \eta) - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' } \biggr]$ $~ \Rightarrow ~~~\mu^2 \equiv \frac{ 2 }{\Chi +1 }$ $~=$ $~ \biggl[ \frac{2 \sinh\eta \cdot \sinh\eta^' }{ \cosh(\eta^' + \eta) - \cos(\theta^' - \theta) } \biggr] \, .$

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

$~\Chi$

$\frac{(\varpi^')^2 + \varpi^2 + (z^' - z)^2}{2\varpi^' \varpi} $

$~ \frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' }$

$~\mu^2 \equiv \frac{2}{\Chi + 1}$

$\frac{4\varpi^' \varpi}{(\varpi^' + \varpi)^2 + (z^' - z)^2}$

$~ \frac{2 \sinh\eta \cdot \sinh\eta^' }{ \cosh(\eta^' + \eta) - \cos(\theta^' - \theta) }$

## Potential

The potential, $~U({\vec{r}}~')$, at a point $~{\vec{r}}~'$ due to an arbitrary mass distribution, $~\rho({\vec{r}})$, is,

 $~U({\vec{r}}~')$ $~=$ $~-G \iiint \frac{\rho(\vec{r}) d^3r}{|~\vec{r} - {\vec{r}}^{~'} ~|} \, .$

See above.

### Green's Function

Wong (1973) points out that in toroidal coordinates the Green's function is,

 $~\frac{1}{|~\vec{r} - {\vec{r}}^{~'} ~|}$ $~=$ $~ \frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 } \sum\limits_{m,n} (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}$ $~ \times \cos[m(\psi - \psi^')][\cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta \end{cases}\, ,$ Wong (1973), Eq. (2.53)

where, $~P^m_{n-1 / 2}, Q^m_{n-1 / 2}$ are "Legendre functions of the first and second kind with order $~n - \tfrac{1}{2}$ and degree $~m$ (toroidal harmonics)," and $~\epsilon_m$ is the Neumann factor, that is, $~\epsilon_0 = 1$ and $~\epsilon_m = 2$ for all $~m \ge 1$. According to CT99, the Green's function written in toroidal coordinates is,

 $~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}$ $~=$ $~ \frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi)$ $~=$ $~ \frac{1}{a\pi} \biggl[ \frac{(\cosh\eta^' - \cos\theta^')}{\sinh\eta^' } \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) \, .$

Things to note:

1. The argument of $~Q_{m - 1 / 2}$ in the CT99 expression is very different from the argument of $~Q^m_{n - 1 / 2}$ (or $~P^m_{n - 1 / 2}$) in Wong's expression.
2. In both expressions, $~m$ is the integer multiplying the azimuthal angle, $~\psi$, but in the CT99 expression this index serves as the subscript index of the function, $~Q$, whereas in Wong's expression it serves as the superscript index of both functions, $~Q$ and $~P$. In this context, note that,
 $~Q^m_{n-\frac{1}{2}}(\cosh\eta)$ $~=$ $~(-1)^m \sqrt{\frac{\pi}{2}} ~\Gamma(m-n+\tfrac{1}{2}) \biggl[ \frac{1}{ \sqrt{\sinh\eta}} \biggr] P^{n}_{m - \frac{1}{2}} (\coth\eta) \, .$
3. Wong's expression contains not only a summation over the index, $~m$, but also an explicit summation over the index, $~n$, which multiplies the "polar" angle, $~\theta$; no such additional summation appears in the CT99 expression, indicating that the summation over $~n$ has implicitly already been completed. In this context, note that the summation expression gives,
 $~ Q^{\mu}_{-\frac{1}{2}}\left(\cosh\xi\right) + 2\sum_{n=1}^{\infty} Q^{\mu}_{n-\frac{1}{2}}\left(\cosh\xi\right) \cos\left[ n (\theta - \theta^') \right]$ $~=$ $~ e^{\mu\pi i} \Gamma\left(\mu+ \tfrac{1}{2} \right) \biggl[ \dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\sinh\xi\right)^{\mu }}{\left\{ \cosh\xi -\cos\left[ n (\theta - \theta^') \right] \right\}^{\mu+(1/2)}}\biggr] \, ;$

or, specifically for the case of $~\mu = 0$,

 $~ \sum_{n=0}^{\infty} \epsilon_n Q_{n-\frac{1}{2}}\left(\cosh\xi\right) \cos\left[ n(\theta - \theta^') \right]$ $~=$ $~ \dfrac{ \pi/\sqrt{2} }{\left[ \cosh\xi-\cos(\theta - \theta^') \right]^{\frac{1}{2}} } \, .$
4. 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,

 $~ Q_\nu[t t^' - (t^2-1)^{1 / 2} (t^{'2} - 1)^{1 / 2} \cos\psi]$ $~=$ $~ Q_\nu(t) P_\nu(t^') + 2\sum_{n=1}^\infty (-1)^n Q^n_\nu(t) P^{-n}_\nu(t^') \cos(n\psi) \, .$

This is valid for,

 $~t, t^'$  real $~1 < t^' < t$ $~\nu \ne -1, -2, -3,$ … $~\psi$   real.

If we make the association, $~t \leftrightarrow \coth\eta$, then we also have,

 $~\frac{1}{\sinh\eta}$ $~=$ $~\sqrt{t^2 - 1} \, ,$

in which case,

 $~ \Chi$ $~=$ $~ \frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^' }$ $~=$ $~ t t^' - (t^2-1)^{1 / 2}(t^{'2}-1)^{1 / 2}\cos(\theta^' - \theta) \, .$

Put together, then, these expressions mean,

 $~ Q_{m - 1 / 2}(\Chi)$ $~=$ $~ Q_{m-1 / 2}(\coth\eta) P_{m - 1 / 2}(\coth\eta^') + 2\sum_{n=1}^\infty (-1)^n Q^n_{m - 1 / 2}(\coth\eta) P^{-n}_{m - 1 / 2}(\coth\eta^') \cos[n(\theta^' - \theta)]$ $~=$ $~ \sum_{n=0}^\infty \epsilon_n (-1)^n Q^n_{m - 1 / 2}(\coth\eta) P^{-n}_{m - 1 / 2}(\coth\eta^') \cos[n(\theta^' - \theta)] \, .$

Also, from our derived $~Q-P$ relation,

 $~Q^m_{n-\frac{1}{2}}(\cosh\eta)$ $~=$ $~ \sqrt{\frac{\pi}{2}} ~\Gamma(n+m + \tfrac{1}{2}) ~(-1)^m\biggl[ \frac{1}{ \sqrt{\sinh\eta}} \biggr] P^{-n}_{m - \frac{1}{2}} (\coth\eta)$ $~\Rightarrow ~~~ P^{-n}_{m - \frac{1}{2}} (\coth\eta)$ $~=$ $~ \sqrt{\frac{2}{\pi}} ~\frac{(-1)^m \sqrt{\sinh\eta} }{\Gamma(n+m + \tfrac{1}{2})} ~ Q^m_{n-\frac{1}{2}}(\cosh\eta) \, .$

we can write,

 $~ Q_{m - 1 / 2}(\Chi)$ $~=$ $~ \sum_{n=0}^\infty \epsilon_n (-1)^n Q^n_{m - 1 / 2}(\coth\eta) \biggl\{ \sqrt{\frac{2}{\pi}} ~\frac{(-1)^m \sqrt{\sinh\eta^'} }{\Gamma(n+m + \tfrac{1}{2})} ~ Q^m_{n-\frac{1}{2}}(\cosh\eta^') \biggr\} \cos[n(\theta^' - \theta)]$

Next, we pull from the accompanying discussion of the Gil et al. (2000) expression,

 $~Q_{n-1 / 2}^m (\lambda)$ $~=$ $~(-1)^n \frac{\pi^{3/2}}{\sqrt{2} \Gamma(n-m+\frac{1}{2})} (x^2-1)^{1 / 4} P_{m-1 / 2}^n(x) \, ,$

where, $~\lambda \equiv x/\sqrt{x^2-1}$. Identifying $~x$ with $~\cosh\eta$, in which case we have $~\lambda = \coth\eta$, and, switching index notation, $~n \leftrightarrow m$, gives,

 $~Q_{m-1 / 2}^n (\coth\eta)$ $~=$ $~(-1)^m \frac{\pi^{3/2}}{\sqrt{2} \Gamma(m-n+\frac{1}{2})} (\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta) \, .$ $~=$ $~ (-1)^n \sqrt{ \frac{\pi}{2} } ~\Gamma(n - m + \tfrac{1}{2} )(\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta) \, .$

where, this last step also incorporates the "Euler reflection formula for gamma functions", namely,

 $~\frac{1}{\Gamma(m-n+\tfrac{1}{2})}$ $~=$ $~\frac{\Gamma(n-m+\frac{1}{2}) }{\pi (-1)^{m+n}} \, .$

So we have,

 $~ Q_{m - 1 / 2}(\Chi)$ $~=$ $~ \sum_{n=0}^\infty \epsilon_n (-1)^n \biggl\{(-1)^n \sqrt{ \frac{\pi}{2} } ~\Gamma(n - m + \tfrac{1}{2} )(\sinh\eta)^{1 / 2} P_{n-1 / 2}^m(\cosh\eta)\biggr\} \biggl\{ \sqrt{\frac{2}{\pi}} ~\frac{(-1)^m \sqrt{\sinh\eta^'} }{\Gamma(n+m + \tfrac{1}{2})} ~ Q^m_{n-\frac{1}{2}}(\cosh\eta^') \biggr\} \cos[n(\theta^' - \theta)]$ $~=$ $~\sqrt{\sinh\eta^'} \sqrt{\sinh\eta} \sum_{n=0}^\infty \epsilon_n (-1)^m \frac{ \Gamma(n - m + \tfrac{1}{2})}{\Gamma(n+m + \tfrac{1}{2})} P_{n-1 / 2}^m(\cosh\eta) Q^m_{n-\frac{1}{2}}(\cosh\eta^') \cos[n(\theta^' - \theta)] \, .$

Hence, the CT99 Green's function may be rewritten as,

 $~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}$ $~=$ $~ \frac{1}{a\pi} [ (\cosh\eta^' - \cos\theta^') (\cosh\eta - \cos\theta)]^{1 / 2} \sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] \sum_{n=0}^\infty \epsilon_n (-1)^m \frac{ \Gamma(n - m + \tfrac{1}{2})}{\Gamma(n+m + \tfrac{1}{2})} P_{n-1 / 2}^m(\cosh\eta) Q^m_{n-\frac{1}{2}}(\cosh\eta^') \cos[n(\theta^' - \theta)]$ $~=$ $~ \frac{1}{a\pi} [ (\cosh\eta^' - \cos\theta^') (\cosh\eta - \cos\theta)]^{1 / 2} \sum_{m=0}^{\infty} \sum_{n=0}^\infty \epsilon_m\epsilon_n (-1)^m \frac{ \Gamma(n - m + \tfrac{1}{2})}{\Gamma(n+m + \tfrac{1}{2})} \cos[m(\psi - \psi^')] \cos[n(\theta^' - \theta)] P_{n-1 / 2}^m(\cosh\eta) Q^m_{n-\frac{1}{2}}(\cosh\eta^') \, .$

Let's compare this with Wong's (1973) Green's function, namely,

 $~\frac{1}{|~\vec{r} - {\vec{r}}^{~'} ~|}$ $~=$ $~ \frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 } \sum\limits_{m,n} (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}$ $~ \times \cos[m(\psi - \psi^')][\cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta \end{cases}\, .$ 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,  $~Q_\nu^\mu(z)$ $~=$ $~ e^{i \mu \pi} ~2^{-\nu - 1} \frac{\Gamma(\nu + \mu + 1) }{\Gamma(\nu + 1) } (z^2 - 1)^{-\mu/2} \int_0^\pi (z+\cos t)^{\mu - \nu - 1} (\sin t)^{2\nu + 1} dt \, .$

This is valid for,

 $~\mathrm{Re} ~\nu > -1$ and $~\mathrm{Re} (\nu + \mu + 1) > 0 \, .$
• Equation (10) in §3.7, p. 156 of Volume I gives,  $~Q_\nu^\mu(z)$ $~=$ $~ e^{i \mu \pi} ~ (2\pi)^{-\frac{1}{2}} (z^2-1)^{\mu/2} ~\Gamma(\mu + \tfrac{1}{2})~\biggl\{ \int_0^\pi (z - \cos t)^{-\mu - \frac{1}{2}} \cos[(\nu + \tfrac{1}{2})t] ~dt -\cos(\nu\pi) \int_0^\infty (z + \cosh t)^{-\mu - \frac{1}{2}} e^{-(\nu + \frac{1}{2})t} ~dt \biggr\}$

This is valid for,

 $~\mathrm{Re} ~\nu > -\tfrac{1}{2}$ and $~\mathrm{Re} (\nu + \mu + 1) > 0 \, .$

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

 $~\int_{-\pi}^{\pi} \frac{\cos[n(\theta - \theta^')] d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}}$ $~=$ $~\int_{-\pi}^{\pi} \frac{ \{\cos(n\theta)\cos(n\theta^') + \sin(n\theta)\sin(n\theta^') \} d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} \, .$

Therefore, we