Editing 2DStructure/ToroidalGreenFunction (section)

==Gravitational Potential==

Quite generally, then, the gravitational potential can be obtained at any coordinate location, <math>~(\eta,\theta,\psi)</math> &#8212; both inside and outside of a specified mass distribution &#8212; by carrying out three nested spatial integrals over the product of:&nbsp; <math>~\rho(\vec{x}^{~'})</math>, the [[#DiffVolumeElement|differential volume element]], and the Green's function as specified ''either'' by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] or by [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)].  

In what follows we will make an effort to elucidate the pros and cons of adopting one Green's function expression over the other.  In each case we begin by writing the expression for the potential in such a way that variations in the azimuthal coordinate, <math>~\psi</math>, are described by ''Fourier components,'' <math>~\Phi_m^{(1)}(\eta,\theta)</math> and <math>~\Phi_m^{(2)}(\eta,\theta)</math>, of the potential, such that,
<div align="center">
<math>~\Phi(\vec{x}) = \tfrac{1}{2}\Phi_0^{(1)}(\eta,\theta) + \sum_{m=1}^\infty \cos (m\psi) \Phi_m^{(1)}(\eta,\theta)  + \sum_{m=1}^\infty \sin (m\psi) \Phi_m^{(2)}(\eta,\theta) \, .</math>
</div>

Each Fourier component of the potential depends explicitly on the corresponding ''Fourier component'' of the density distribution, defined such that,  
<div align="center">
<math>~\rho(\vec{x}) = \tfrac{1}{2}\rho_0^{(1)}(\eta,\theta) + \sum_{m=1}^\infty \cos (m\psi) \rho_m^{(1)}(\eta,\theta)  + \sum_{m=1}^\infty \sin (m\psi) \rho_m^{(2)}(\eta,\theta) \, .</math>
</div>

<table border="1" align="center" cellpadding="8" width="70%">
<tr>
  <th align="center" bgcolor="yellow">
LaTeX mathematical expressions cut-and-pasted directly from
<br />
NIST's ''Digital Library of Mathematical Functions''
  </th>
</tr>
<tr>
  <td align="left">
As an additional primary point of reference, note that according to [https://dlmf.nist.gov/1.8 &sect;1.8(i) of NIST's ''Digital Library of Mathematical Functions''], a Fourier Series is defined as follows:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{2}a_{0}+\sum^{\infty}_{n=1}\biggl[ a_{n}\cos\bigl(nx\bigr)+b_{n}\sin\bigl(nx\bigr) \biggr],</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~a_n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\cos\bigl(nx\bigr)\mathrm{d}x,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\pi}\int^{\pi}_{-\pi}f(x)\sin\bigl(nx\bigr)\mathrm{d}x.</math>
  </td>
</tr>
</table>

  </td>
</tr>
</table>

Notice, therefore, that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho_m^{(1)}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\pi}\int^{\pi}_{-\pi}\rho(\eta,\theta,\psi)\cos\bigl(m\psi\bigr)\mathrm{d}\psi,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~\rho_m^{(2)}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\pi}\int^{\pi}_{-\pi}\rho(\eta,\theta,\psi)\sin\bigl(m\psi\bigr)\mathrm{d}\psi \, .</math>
  </td>
</tr>
</table>


===The CT99 Expression for the Potential===
====In Three-Dimensional Generality====
Employing the Green's function expression derived by [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], the gravitational potential for any three-dimensional matter distribution is,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi(\eta,\theta,\psi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -G
\iiint \rho(\eta^',\theta^',\psi^') \biggl\{ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr\}  \biggl[ \frac{a^3 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] d\eta^'~ d\theta^'~ d\psi^'
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\frac{Ga^2}{\pi} \int d\eta^' \int d\theta^' \int d\psi^'
\iiint \rho(\eta^',\theta^',\psi^')  \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ \times
\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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\frac{Ga^2}{\pi} \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \sum_{m=0}^{\infty} \epsilon_m
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi)
\int d\psi^' \rho(\eta^',\theta^',\psi^')  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ \times
[\cos(m\psi)\cos(m\psi^') + \sin(m\psi)\sin(m\psi^') ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\frac{Ga^2}{\pi} \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \biggl\{
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- \frac{1}{2}}(\Chi)
\int_{-\pi}^{\pi} d\psi^' \rho(\eta^',\theta^',\psi^')  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ + \sum_{m=1}^{\infty} 2\cos(m\psi)
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi)
\int_{-\pi}^\pi d\psi^' \rho(\eta^',\theta^',\psi^')  \cos(m\psi^')
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ + \sum_{m=1}^{\infty} 2\sin(m\psi)
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi)
\int_{-\pi}^\pi d\psi^' \rho(\eta^',\theta^',\psi^')  \sin(m\psi^')
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - Ga^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} \biggl\{
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- \frac{1}{2}}(\Chi)  \rho_0^{(1)}(\eta^',\theta^')
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ + \sum_{m=1}^{\infty} 2\cos(m\psi)
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi) \rho_m^{(1)}(\eta^',\theta^')
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ + \sum_{m=1}^{\infty} 2\sin(m\psi)
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi) \rho_m^{(2)}(\eta^',\theta^')
\biggr\} \, .
</math>
  </td>
</tr>
</table>

We conclude, therefore, that each one of the Fourier components of the gravitational potential is given by the expression,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~\Phi_m^{(1),(2)}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 2Ga^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} 
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{m- 1 / 2}(\Chi) \rho_m^{(1),(2)}(\eta^',\theta^') \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], p. 88, Eq. (20)
  </td>
</tr>
</table>
where, [[#As_Presented_in_Cohl_.26_Tohline_.281999.29|as above]],
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Chi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^'} \, .
</math>
  </td>
</tr>
</table>

====For Axisymmetric Systems====
For axisymmetric systems, the density distribution has no dependence on the azimuthal coordinate, <math>~\psi</math>.  Hence, for all <math>~m > 0</math>, the ''Fourier components'' of the density, <math>~\rho_m^{(1),(2)}</math>, are zero.  The only nonzero component is, <math>~\rho_0^{(1)}(\eta,\theta) = 2\rho(\eta,\theta)</math>.  For axisymmetric systems, then, the gravitational potential is,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{2}\Phi_0^{(1)}(\eta,\theta)</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 2Ga^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} 
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- 1 / 2}(\Chi) \rho(\eta^',\theta^') \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], p. 88, Eqs. (31) &amp; (32a)
  </td>
</tr>
</table>

===Wong's Expression for the Potential===

====Fully Three-Dimensional Case====

Employing [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] Green's function expression, the gravitational potential for any three-dimensional matter distribution is,
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi(\vec{x})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -G
\iiint \rho(\eta^',\theta^',\psi^') \biggl\{ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr\}  \biggl[ \frac{a^3 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] d\eta^'~ d\theta^'~ d\psi^'
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{a^2G}{\pi} \int d\eta^' \int d\theta^' \int d\psi^'
\biggl[ \frac{\rho(\eta^',\theta^',\psi^') ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] 
\biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 }
\sum\limits^\infty_{n=0} \sum\limits^\infty_{m=0}(-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \cos[m(\psi - \psi^')]\cos[n(\theta - \theta^')] ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{a^2G}{\pi} (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \sum\limits^\infty_{m=0}(-1)^m \epsilon_m ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
\int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times~
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\int d\psi^' \rho(\eta^',\theta^',\psi^') [\cos(m\psi)\cos(m\psi^') + \sin(m\psi) \sin(m\psi^')]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{a^2G}{\pi} (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \biggl\{
\int d\eta^' ~P_{n-1 / 2}(\cosh\eta_<) ~Q_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\int_{-\pi}^\pi d\psi^' \rho(\eta^',\theta^',\psi^') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~
\sum\limits^\infty_{m=1} 2\cos(m\psi)(-1)^m  ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
\int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\int_{-\pi}^\pi d\psi^' \rho(\eta^',\theta^',\psi^') \cos(m\psi^') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~
\sum\limits^\infty_{m=1} 2 \sin(m\psi)(-1)^m ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
\int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\int_{-\pi}^\pi  d\psi^' \rho(\eta^',\theta^',\psi^') \sin(m\psi^')
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - a^2G (\cosh\eta - \cos\theta)^{1 / 2} \sum\limits^\infty_{n=0} \epsilon_n \biggl\{
\int d\eta^' ~P_{n-1 / 2}(\cosh\eta_<) ~Q_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\rho_0^{(1)}(\eta^',\theta^') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~
\sum\limits^\infty_{m=1} 2\cos(m\psi)(-1)^m  ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
\int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\rho_m^{(1)}(\eta^',\theta^')
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~
\sum\limits^\infty_{m=1} 2 \sin(m\psi)(-1)^m ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
\int d\eta^' ~P^m_{n-1 / 2}(\cosh\eta_<) ~Q^m_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl[ \frac{ ~\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr]\cos[n(\theta - \theta^')]
\rho_m^{(2)} (\eta^',\theta^')
\biggr\} \, .
</math>
  </td>
</tr>
</table>

We conclude, therefore, that each one of the Fourier components of the gravitational potential is given by the expression,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~\Phi_m^{(1),(2)} (\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2Ga^2 (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n (-1)^m  ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\int d\eta^' ~\sinh\eta^' ~ P^m_{n-1 / 2}(\cosh\eta_<) ~ Q^m_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl\{ \frac{ \cos[n(\theta - \theta^')]}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr\} 
\rho_m^{(1),(2)}(\eta^',\theta^') \, .
</math>
  </td>
</tr>
</table>

====Axisymmetric Systems====
For axisymmetric systems, the density distribution has no dependence on the azimuthal coordinate, <math>~\psi</math>.  Hence, for all <math>~m > 0</math>, the ''Fourier components'' of the density, <math>~\rho_m^{(1),(2)}</math>, are zero.  The only nonzero component is, <math>~\rho_0^{(1)}(\eta,\theta) = 2\rho(\eta,\theta)</math>.  For axisymmetric systems, then, the gravitational potential is,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{2}\Phi_0^{(1)}(\eta,\theta)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2Ga^2 (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n 
\int d\eta^' ~\sinh\eta^'~P^0_{n-1 / 2}(\cosh\eta_<) ~Q^0_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl\{ \frac{ ~\cos[n(\theta - \theta^')]}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr\}
\rho(\eta^',\theta^') \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], p. 293, Eq. (2.55)
  </td>
</tr>
</table>