Editing AxisymmetricConfigurations/PoissonEq (section)

=Special Functions &amp; Other Broadly Used Representations=

==Spherical Harmonics and Associated Legendre Functions==


<div align="center" id="Ylm">
<table border="1" cellpadding="8" align="center" width="80%">
<tr>
<th align="center" colspan="2"><font size="+0">Table 2: <br />Green's Function in Terms of Associated Legendre Functions, 
<math>~P_\ell^m(\cos\theta)</math>, <br />and the Spherical Harmonics, <math>~Y_{\ell m}(\theta,\phi)</math></font></th>
</tr>
<tr>
  <td align="left" colspan="2">
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'} |}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{4\pi}{2\ell+1} \biggl[  \frac{r_<^\ell}{r_>^{\ell+1}} \biggr] Y_{\ell m}^*(\theta^', \phi^') Y_{\ell m}(\theta,\phi)</math>
  </td>
</tr>
</table>
<div align="center">
[https://archive.org/details/ClassicalElectrodynamics2nd J. D. Jackson (1975, 2<sup>nd</sup> Edition)], Eq. (3.70)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 656, Eq. (1C-31)
</div>
  </td>

</tr>
<tr>
  <td align="left" rowspan="2">
Note:<br />
<div align="center"><math>~Y_{\ell m}(\theta,\phi) = \biggl[ \frac{(2\ell + 1 )(\ell - m)!}{4\pi(\ell + m)!} \biggr]^{1 / 2} P_\ell^m(\cos\theta)e^{im\phi}</math><br /><br />
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], Eq. (3.53)<br />
[https://dlmf.nist.gov/14.30#i DLMF] &sect;14.30i<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 655, Eq. (1C-27)<br /><br />
<math>~Y_{\ell,-m}(\theta,\phi) = (-1)^{m}Y_{\ell m}^*(\theta,\phi)</math><br />
</div>


----


<div align="center"><math>~P_{\ell}^m(x) = (-1)^m (1-x^2)^{m/2} ~ \frac{d^m}{dx^m} P_\ell(x)</math><br />
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], Eq. (3.49)<br />
[https://dlmf.nist.gov/14.6#i DLMF] &sect;14.6i<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 655, Eq. (1C-20)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], p. 481, Eq. (C.5)
</div>


----


<div align="center"><math>~P_{\ell}(x) = \frac{1}{2^\ell \ell !} ~ \frac{d^\ell}{dx^\ell} (x^2 - 1)^\ell</math><br />
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], Eq. (3.16)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 655, Eq. (1C-21)
</div>
  </td>
  <th align="center">Leading Legendre Functions
<div align="center">
[https://archive.org/details/FieldTheoryHandbookMoonAndSpencer P. Moon &amp; D. E. Spencer (1971)], p. 205<br />
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], Eq. (3.15)<br />
[https://dlmf.nist.gov/18.5#iv DLMF] &sect;18.5iv<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 655, Eq. (1C-25)
</div>
  </th>
</tr>
<tr>
  <td align="center" rowspan="1">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~P_0(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~P_1(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~P_2(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{2}(3x^2 - 1)</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~P_3(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{2}(5x^3 - 3x)</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~P_4(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{8}(35x^4 - 30x^2 + 3)</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~P_5(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{8}(63x^5 - 70x^3 + 15x)</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~P_6(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{16}(231x^6 - 315x^4 + 105x^2 - 5)</math>
  </td>
</tr>
<tr>
<td align="center" colspan="3">
[http://astrowww.phys.uvic.ca/~tatum/celmechs/celm5.pdf J. B. Tatum (2021)] Celestial Mechanics class notes (UVic), &sect;5.11, p. 53, Eq. (5.11.7)
</td>
</tr>
</table>
  </td>
</tr>

<tr>
  <th align="center" colspan="2">Leading Spherical Harmonics
<div align="center">
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], &sect;3.5<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 656, Eq. (1C-33)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], p. 482, Eqs. (C.7)  - (C.16)
</div>
  </th>
</tr>
<tr>
  <td align="left" rowspan="1">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~Y_{00} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\sqrt{4\pi}}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~Y_{1,\pm 1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mp \sqrt{\frac{3}{8\pi}} ~\sin\theta ~e^{\pm i\phi}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~Y_{10} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{\frac{3}{4\pi}} ~\cos\theta </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~Y_{2,\pm 2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4} \sqrt{\frac{15}{2\pi}} ~\sin^2\theta ~e^{\pm 2i\phi}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~Y_{2, \pm 1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mp \sqrt{\frac{15}{8\pi}} ~\sin\theta \cos\theta ~e^{\pm i\phi}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~Y_{20} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{\frac{5}{4\pi}} ~(\tfrac{3}{2}\cos^2\theta - \tfrac{1}{2})</math>
  </td>
</tr>
</table>

  </td>
  <td align="left" rowspan="1">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~Y_{3,\pm 3} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mp \frac{1}{4} \sqrt{\frac{35}{4\pi}} ~\sin^3\theta  ~e^{\pm 3i\phi}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~Y_{3, \pm 2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4}\sqrt{\frac{105}{2\pi}} ~\sin^2\theta \cos\theta ~e^{\pm 2i\phi}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~Y_{3, \pm 1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mp \frac{1}{4}\sqrt{\frac{21}{4\pi}} ~\sin\theta (5\cos^2\theta - 1) ~e^{\pm i\phi}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~Y_{30} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{\frac{7}{4\pi}} ~(\tfrac{5}{2}\cos^3\theta - \tfrac{3}{2}\cos\theta) </math>
  </td>
</tr>
</table>

----


Note that, for all cases where <math>~m=0</math> :<br />
<div align="center"><math>~Y_{\ell 0}(\theta,\phi) = \sqrt{\frac{(2\ell + 1 )}{4\pi} } ~P_\ell(\cos\theta)</math><br />
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], Eq. (3.57)
</div>
  </td>
</tr>
</table>
</div>

==Multipole Expansions==

===Mass Multipole Moments===

As an extension of the [[#MultipoleMoments|above discussion of]]
<table border="0" cellpadding="5" align="center">

<tr>
<td align="center" colspan="3">
<font color="#770000">'''Multipole Moments of the Mass Distribution'''</font>
</td>
</tr>
<tr>
  <td align="right">
<math>~q_{\ell m}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\int (r^')^\ell  Y_{\ell m}^*(\theta^', \phi^')
~\rho(r^', \theta^', \phi^') d^3x^' 
\, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], p. 137, Eq. (4.3)
  </td>
</tr>
</table>
here we evaluate a set of the leading order mass-multipole moments in terms of their cartesian-coordinate representations.

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

<tr>
  <td align="right">
<math>~q_{00}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\sqrt{4\pi}}
\int \rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{11}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int r^'  \biggl[ Y_{11}^* \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- ~\int r^'  \biggl[ \sqrt{\frac{3}{8\pi}}\sin\theta ~\biggl(\cos\phi^' - i\sin\phi^' \biggr) \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \sqrt{\frac{3}{8\pi}}~\int r^'  \biggl[ \frac{\sqrt{(x^')^2+(y^')^2}}{r^'} 
~\biggl(\frac{x^'}{\sqrt{(x^')^2+(y^')^2}} - \frac{iy^'}{\sqrt{(x^')^2+(y^')^2}} \biggr)\biggr]
~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \sqrt{\frac{3}{8\pi}}~\int (x^' - iy^')
~\rho(\vec{x}^{~'}) d^3x^' 
\, ;
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~q_{1,- 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+ \sqrt{\frac{3}{8\pi}}~\int (x^' + iy^')
~\rho(\vec{x}^{~'}) d^3x^' 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{10}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int r^'  \biggl[ Y_{10}^* \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int r^'  \biggl[ \sqrt{\frac{3}{4\pi}}~\cos\theta^' \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sqrt{\frac{3}{4\pi}} \int z^' \rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{22}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int (r^')^2  \biggl[ Y_{22}^*(\theta^', \phi^') \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int (r^')^2  \biggl\{ \frac{1}{4}\sqrt{\frac{15}{2\pi}} ~\sin^2\theta^' ~\biggl[\cos(2\phi^') - i\sin(2\phi^')\biggr] \biggr\}~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4}\sqrt{\frac{15}{2\pi}} 
\int (r^')^2  \biggl\{ \frac{(x^')^2+(y^')^2}{(r^')^2}  ~\biggl[\frac{ (x^')^2 - (y^')^2 - 
2i ( x^' y^' )}{(x^')^2+(y^')^2} \biggr] \biggr\}~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4}\sqrt{\frac{15}{2\pi}} 
\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~q_{2,-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4}\sqrt{\frac{15}{2\pi}} 
\int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{21}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int (r^')^2  \biggl[ Y_{21}^*(\theta^', \phi^') \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \sqrt{\frac{15}{8\pi}} \int (r^')^2  \biggl[ \sin\theta^' \cos\theta^' (\cos\phi^' -i\sin\phi^') \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \sqrt{\frac{15}{8\pi}} \int (r^')^2  
\biggl[ \frac{z^' \sqrt{(x^')^2+(y^')^2}}{(r^')^2} 
~\biggl(\frac{x^'}{\sqrt{(x^')^2+(y^')^2}} - \frac{iy^'}{\sqrt{(x^')^2+(y^')^2}} \biggr)\biggr]
~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \sqrt{\frac{15}{8\pi}} \int z^'(x^' - iy^')  
~\rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~q_{2,-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+ \sqrt{\frac{15}{8\pi}} \int z^'(x^' + iy^')  
~\rho(\vec{x}^{~'}) d^3x^' \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{20}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int (r^')^2  \biggl[ Y_{20}^*(\theta^', \phi^') \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int (r^')^2  \biggl[ \sqrt{\frac{5}{16\pi}} (3\cos^2\theta^' - 1) \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{\frac{5}{16\pi}} 
\int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \, .
</math>
  </td>
</tr>
</table>
</div>


<div align="center" id="qlm">
<table border="1" align="center" cellpadding="8" width="80%">
<tr><th>Table 3: &nbsp; Summary of Cartesian Expressions for Leading Mass-Multipole Moments</th></tr>

<tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~q_{0, 0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\sqrt{4\pi}}
\int \rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{1,\pm 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mp \sqrt{\frac{3}{8\pi}}~\int (x^' \mp iy^')
~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{1,0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sqrt{\frac{3}{4\pi}} \int z^' \rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{2,\pm 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4}\sqrt{\frac{15}{2\pi}} 
\int (x^' \mp iy^')^2~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{2,\pm 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mp \sqrt{\frac{15}{8\pi}} \int z^'(x^' \mp iy^')  
~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~q_{2,0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{\frac{5}{16\pi}} 
\int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

</table>

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

===Contributions to the Gravitational Potential in Terms of Multipole Moments===

Expanding on the [[#PhiSeriesExpansion|above discussion of the gravitational potential, written as a series expansion in terms of mass-multipole moments and spherical harmonics]],
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(r,\theta,\phi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \Phi_B(r,\theta,\phi) \biggr|_{\ell=0} +~ \Phi_B(r,\theta,\phi)\biggr|_{\ell=1} + ~\Phi_B(r,\theta,\phi)\biggr|_{\ell=2} 
-~4\pi G  
\sum_{\ell=3}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{q_{\ell m}}{r^{\ell+1}} \frac{Y_{\ell m}(\theta,\phi)}{(2\ell+1)}  \, ,
</math>
  </td>
</tr>
</table>
here we evaluate the first three, leading order terms.

====First Term====
First (term with <math>~\ell = 0</math>):
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(r,\theta,\phi)\biggr|_{\ell = 0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-4\pi G  
\biggl[ q_{00}  \biggr] \frac{ Y_{00}(\theta,\phi)}{r} 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-4\pi G  
\biggl[ \frac{1}{\sqrt{4\pi}}
\int \rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{1}{\sqrt{4\pi}~r}  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{GM}{r} \, .  
</math>
  </td>
</tr>
</table>

====Second Term====
Second (terms with <math>~\ell = 1</math>):
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(r,\theta,\phi)\biggr|_{\ell = 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4\pi G}{3}  
\sum_{m=-1}^{+1} \biggl[ q_{1 m}\biggr] \frac{Y_{1 m}(\theta,\phi)}{r^2}  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4\pi G}{3r^2}  
\biggl\{ 
\biggl[ q_{1,- 1}\biggr] Y_{1, -1}(\theta,\phi) 
+ \biggl[ q_{1 0}\biggr] Y_{1 0}(\theta,\phi) 
+ \biggl[ q_{1 1}\biggr] Y_{1 1}(\theta,\phi) 
\biggr\} 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4\pi G}{3r^2}  
\biggl\{ 
\biggl[ + \frac{3}{8\pi}~\int (x^' + iy^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{(x - iy) }{r}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl[ \frac{3}{4\pi} \int z^' \rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{z}{r} 
+ \biggl[ \frac{3}{8\pi}~\int (x^' - iy^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{(x + iy) }{r}
\biggr\} 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{r^3}  
\biggl\{ 
\biggl[ \frac{1}{2}~\int (x^' + iy^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] (x - iy) 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl[ \int z^' \rho(\vec{x}^{~'}) d^3x^' \biggr] z
+ \biggl[ \frac{1}{2}~\int (x^' - iy^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] (x + iy)
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{r^3}  
\biggl\{ 
\biggl[ \int (x^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] (x ) 
+ \biggl[ \int (y^')~\rho(\vec{x}^{~'}) d^3x^' \biggr] (y ) 
+ \biggl[ \int z^' \rho(\vec{x}^{~'}) d^3x^' \biggr] z
\biggr\} 
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{GM}{r^3} \biggl[ \vec{x} \cdot \vec{x}_\mathrm{com} \biggr]</math>
  </td>
</tr>
<tr>
  <td align="right">
[https://en.wikipedia.org/wiki/Center_of_mass#A_continuous_volume where]:
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\vec{x}_\mathrm{com} \equiv \frac{1}{M} \int ~\vec{x}^{~'} \rho(\vec{x}^{~'}) d^3x^'\, .
</math>
  </td>
</tr>
</table>

====Third Term====

Third (terms with <math>~\ell = 2</math>):
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(r,\theta,\phi)\biggr|_{\ell = 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4\pi G}{5r^3} \biggl\{
\biggl[ q_{2, -2} \biggr] Y_{2,-2}  + \biggl[ q_{2, 2} \biggr] Y_{2,2}  
+ \biggl[ q_{2,-1} \biggr] Y_{2, -1} + \biggl[ q_{2,1} \biggr] Y_{2, 1} 
+ \biggl[ q_{20} \biggr] Y_{20} \biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4\pi G}{5r^3} \biggl\{
\biggl[ \frac{1}{4}\sqrt{\frac{15}{2\pi}} 
\int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{1}{4}\sqrt{\frac{15}{2\pi}}  \sin^2\theta e^{-2i\phi}
+ \biggl[\frac{1}{4}\sqrt{\frac{15}{2\pi}} 
\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \frac{1}{4}\sqrt{\frac{15}{2\pi}}   \sin^2\theta e^{2i\phi}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl[ \sqrt{\frac{15}{8\pi}} \int z^'(x^' + iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  \sqrt{\frac{15}{8\pi}} \sin\theta \cos\theta e^{-i\phi}
+ \biggl[ \sqrt{\frac{15}{8\pi}} \int z^'(x^' - iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  \sqrt{\frac{15}{8\pi}} \sin\theta \cos\theta e^{i\phi}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl[ \sqrt{\frac{5}{16\pi}} 
\int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr] \sqrt{\frac{5}{16\pi}} (3\cos^2\theta - 1)  
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4\pi G}{5r^3} \biggl\{
\frac{3\cdot 5}{2^5 \pi} \biggl(\frac{\varpi}{r}\biggr)^2
\biggl[ \int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ \cos(2\phi) - i\sin(2\phi) \biggr]
+ \frac{3\cdot 5}{2^5 \pi} \biggl(\frac{\varpi}{r}\biggr)^2
\biggl[\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ \cos(2\phi) + i\sin(2\phi) \biggr]
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  \frac{3\cdot 5}{2^3\pi} \biggl( \frac{\varpi z}{r^2} \biggr) \biggl[\int z^'(x^' + iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  \biggl[ \cos\phi - i\sin\phi \biggr]
+ \frac{3\cdot 5}{2^3\pi} \biggl( \frac{\varpi z}{r^2} \biggr) \biggl[ \int z^'(x^' - iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]   \biggl[ \cos\phi + i\sin\phi \biggr]
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \frac{5}{2^4\pi}  \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]  \biggl( \frac{3z^2 - r^2}{r^2} \biggr)  
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{4G}{r^5} \biggl\{
\frac{3}{2^5 } 
\biggl[ \int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ x^2 - y^2 - 2i xy\biggr]
+ \frac{3}{2^5 } 
\biggl[\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] \biggl[ x^2 - y^2 + 2i xy \biggr]
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  \frac{3z}{2^3}   \biggl[\int z^'(x^' + iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  \biggl[x - i y \biggr]
+ \frac{3z}{2^3} \biggl[ \int z^'(x^' - iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]   \biggl[x + i y \biggr]
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  \biggl( \frac{3z^2 - r^2}{2^4} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
\frac{3( x^2 - y^2 )}{2^2 } 
\biggl[ \int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' + \int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+ \frac{3i xy}{2 } 
\biggl[\int (x^' -iy^')^2~\rho(\vec{x}^{~'}) d^3x^' -  \int (x^' + iy^')^2~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  3xz  \biggl[\int z^'(x^' + iy^')  ~\rho(\vec{x}^{~'}) d^3x^' + \int z^'(x^' - iy^')  ~\rho(\vec{x}^{~'}) d^3x^'\biggr]  
+ 3i yz\biggl[ \int z^'(x^' - iy^')  ~\rho(\vec{x}^{~'}) d^3x^' - \int z^'(x^' + iy^')  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  \biggl( \frac{3z^2 - r^2}{2} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
\frac{3( x^2 - y^2 )}{2^2 } 
\biggl[ \int [2(x^')^2 -  2(y^')^2]~\rho(\vec{x}^{~'}) d^3x^'  \biggr] 
+ \frac{3i xy}{2 } 
\biggl[\int -4i x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  3xz  \biggl[\int 2x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 3i yz\biggl[ \int -2iy^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+  \biggl( \frac{3z^2 - r^2}{2} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
\frac{3( x^2 - y^2 )}{2} 
\biggl[ \int [(x^')^2 -  (y^')^2]~\rho(\vec{x}^{~'}) d^3x^'  \biggr] 
+ 6 xy \biggl[\int x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+  6xz  \biggl[\int x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 6yz\biggl[ \int y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+  \biggl( \frac{3z^2 - r^2}{2} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}  \, .
</math>
  </td>
</tr>
</table>

Rearranging terms, we have,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(r,\theta,\phi)\biggr|_{\ell = 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+  2xz  \biggl[\int 3x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{3( x^2 - y^2 )}{2} 
\biggl[ \int [(x^')^2 -  (y^')^2]~\rho(\vec{x}^{~'}) d^3x^'  \biggr] 
+ \biggl( \frac{3z^2 - r^2}{2} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+  2xz  \biggl[\int 3x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{3( x^2  )}{2} 
\biggl[ \int [(x^')^2 -  (y^')^2]~\rho(\vec{x}^{~'}) d^3x^'  \biggr] 
-  \biggl( \frac{x^2}{2} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
-\frac{3( y^2 )}{2} 
\biggl[ \int [(x^')^2 -  (y^')^2]~\rho(\vec{x}^{~'}) d^3x^'  \biggr] 
- \biggl( \frac{y^2}{2} \biggr) \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ z^2 \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+  2xz  \biggl[\int 3x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ z^2 \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{( x^2  )}{2} 
\biggl[ \int [3(x^')^2 -  3(y^')^2-3(z^')^2 + (r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{( y^2 )}{2} 
\biggl[ \int [3(x^')^2 -  3(y^')^2 -3(z^')^2 + (r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+  2xz  \biggl[\int 3x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ z^2 \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{( x^2  )}{2} 
\biggl[ \int [3(x^')^2 -  3(y^')^2-3(z^')^2 + 3(r^')^2 - 3(r^')^2 + (r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{( y^2 )}{2} 
\biggl[ \int [-3(x^')^2 +  3(y^')^2 -3(z^')^2 + 3(r^')^2 - 3(r^')^2 + (r^')^2 ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+  2xz  \biggl[\int 3x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ z^2 \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
+\frac{( x^2  )}{2} 
\biggl[ \int [6(x^')^2  - 2(r^')^2  ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\frac{( y^2 )}{2} 
\biggl[ \int [6(y^')^2  - 2(r^')^2  ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \biggl\{
2 xy \biggl[\int 3x^' y^'~\rho(\vec{x}^{~'}) d^3x^' \biggr] 
+  2xz  \biggl[\int 3x^' z^'  ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
+ 2yz\biggl[ \int 3y^' z^' ~\rho(\vec{x}^{~'}) d^3x^' \biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ x^2
\biggl[ \int [3(x^')^2  - (r^')^2  ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
+ y^2
\biggl[ \int [3(y^')^2  - 2(r^')^2  ]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
+ z^2 \biggl[  \int \biggl[ 3(z^')^2 - (r^')^2 \biggr]~\rho(\vec{x}^{~'}) d^3x^' \biggr]   
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{G}{2r^5} \sum_{i=1}^3 \sum_{j=1}^3 Q_{i,j} \biggl[ x_i x_j \biggr] \, ,
</math>
  </td>
</tr>
</table>
<span id="QuadrupoleMomentTensor">where,</span> following [https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], we have introduced the,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="center" colspan="3">
<font color="#770000">'''Traceless Quadrupole Moment Tensor'''</font>
</td>
</tr>
<tr>
  <td align="right">
<math>~Q_{i,j}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\int [ 3(x_i^') (x_j^') - (r^')^2 \delta_{ij} ] \rho(\vec{x}^{~'}) d^3x^' \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], p. 138, Eq. (4.9)<br />
[http://adsabs.harvard.edu/abs/1978ApJ...224..497N M. L. Norman &amp; J. R. Wilson (1978)], p. 501, Eq. (19)<br />
  </td>
</tr>
</table>

==Familiar Expression for the Cylindrical Green's Function Expansion==
<div align="center" id="CylindricalGreenFunction">
<table border="1" cellpadding="8" align="center" width="80%">
<tr>
<th align="center"><font size="+0">Table 4: &nbsp;Cylindrical-Coordinate Green's Function Expansion</font></th>
</tr>
<tr>
  <td align="left">
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\sum_{m=-\infty}^{\infty} e^{im(\phi - \phi^')} \int_0^\infty dk ~J_m(k\varpi) ~J_m(k\varpi^') e^{-k(z_> - z_<)}
</math>
  </td>
</tr>
</table>
where <math>~J_m</math> is an order <math>~m</math> Bessel function of the first kind.<br />
<div align="center">
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], exercise [3.14]<br />
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl &amp; J. E. Tohline (1999)], p. 88, Eq. (3)
</div>
  </td>
</tr>
</table>
</div>

==Toroidal Functions==

<!-- NEW TABLE 3 -->
<div align="center" id="Toroidal">
<table border="1" cellpadding="8" align="center" width="80%">
<tr>
<th align="center"><font size="+0">Table 5: &nbsp;Green's Function in Terms of<br />Zero Order, Half-(Odd)Integer Degree, Associated Legendre Functions of the Second Kind, <math>~Q^0_{m-1 / 2}(\chi)</math><br />(also referred to as Toroidal Functions)</font></th>
</tr>
<tr>
  <td align="left">
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=-\infty}^{\infty} e^{im(\phi - \phi^')}Q_{m- 1 / 2}(\chi)
</math>
  </td>
</tr>
</table>
where:<br />
<div align="center"><math>~\chi \equiv \frac{\varpi^2 + (\varpi^')^2 + (z - z^')^2}{2\varpi \varpi^'}</math><br /><br />
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl &amp; J. E. Tohline (1999)], p. 88, Eqs. (15) &amp; (16)<br />
See also the [https://dlmf.nist.gov/14.19#ii DLMF's definition of Toroidal Functions], <math>~Q_{m - 1 / 2}^{0}</math>
</div>
  </td>
</tr>

<tr>
  <td align="left">
Note that, according to, for example, equation (8.731.5) of Gradshteyn &amp; Ryzhik (1994), 
<div align="center">
<math>~Q^0_{-m - 1 / 2}(\chi) = Q^0_{m- 1 / 2}(\chi) \, .</math>
</div>
Hence, the Green's function can straightforwardly be rewritten in terms of a simpler summation over just ''non-negative'' values of the index, <math>~m</math>.
  </td>
</tr>

<tr>
  <td align="left">
Referencing equations (8.13.3) and (8.13.7), respectively, of Abramowitz &amp; Stegun (1965), we see that for the smallest two values of the ''non-negative'' index, <math>~m</math>, the function, <math>~Q_{m- 1 / 2}(\chi)</math>, can be rewritten in terms of, the more familiar, complete elliptic integrals of the first and second kind.  Specifically, 
<table border="0" cellpadding="1" align="center" width="100%">

<tr>
  <td align="left" colspan="3">
for <math>~m = 0</math>,
  </td>
</tr>
<tr>
  <td align="right">
<math>~Q_{m- 1 / 2}(\chi) ~\rightarrow~ Q_{-1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mu K(\mu) \, ,
</math>
  </td>
</tr>
<tr>
  <td align="left" colspan="3">
and, for <math>~m = 1</math>,
  </td>
</tr>

<tr>
  <td align="right">
<math>~Q_{m- 1 / 2}(\chi) ~\rightarrow~ Q_{+ 1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\chi \mu K(\mu) - (1+\chi)\mu E(\mu) \, ,
</math>
  </td>
</tr>
<tr>
  <td align="left" colspan="3">
where,
  </td>
</tr>
<tr>
  <td align="right">
<math>~\mu \equiv \biggl[ \frac{2}{1+\chi} \biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{4\varpi \varpi^'}{(\varpi + \varpi^')^2 + (z - z^')^2} \biggr]^{1 / 2}
\, .
</math>
  </td>
</tr>
</table>
----

<div align="center">
Excerpt from p. 337 of [https://books.google.com/books?id=MtU8uP7XMvoC&printsec=frontcover&dq=Abramowitz+and+stegun&hl=en&sa=X&ved=0ahUKEwialra5xNbaAhWKna0KHcLAASAQ6AEILDAA#v=onepage&q=Abramowitz%20and%20stegun&f=false M. Abramowitz &amp; I. A. Stegun (1995)]
<!--, ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables''-->
</div>
<!-- [[File:AbramowitzStegun ToroidalFunctions2.png|center|700px|Abramowitz &amp; Stegun (1965)]] -->
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="left"><b>&sect;8.13.3</b></td>
  <td align="right">
<math>Q_{-\tfrac12}(z)</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\sqrt{\frac{2}{z+1}} K\biggl( \sqrt{\frac{2}{z+1}} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="left"><b>&sect;8.13.7</b></td>
  <td align="right">
<math>Q_{\tfrac12}(z)</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
z\sqrt{\frac{2}{z+1}} K\biggl( \sqrt{\frac{2}{z+1}} \biggr) - [ 2(z+1)]^{1 / 2} E\biggl( \sqrt{\frac{2}{z+1}} \biggr) 
</math>
  </td>
</tr>
</table>
  </td>
</tr>

<tr>
  <td align="left">
Finally, equation (8.5.3) from Abramowitz &amp; Stegun (1965) or equation (8.832.4) of Gradshteyn &amp; Ryzhik (1994) &#8212; also see equation (2) of [http://adsabs.harvard.edu/abs/2000JCoPh.161..204G Gil, Segura &amp; Temme (2000)] &#8212; provide the recurrence relation for all other values of the index, <math>~m</math>.  Specifically, for all <math>~m \ge 2</math>,
<div align="center">
<math>~Q_{m - 1 / 2}(\chi) = 4\biggl[\frac{m-1}{2m-1}\biggr] \chi Q_{m- 3 / 2}(\chi) 
- \biggl[ \frac{2m-3}{2m-1}\biggr] Q_{m- 5 / 2}(\chi) \, .</math>
</div>
----

<div align="center">
Excerpt from p. 490 of [https://dl-acm-org.libezp.lib.lsu.edu/citation.cfm?id=365474&picked=prox W. Guatschi (1965, Communications of the ACM, vol. 8, issue 8, 488 - 492)]
</div>
<!-- [[File:ToroidalRecurrenceRelation.png|center|500px|Guatschi (1965, Communications of the ACM, 8, 488 - 492)]] -->
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="left"><b>procedure</b> ''toroidal'';</td>
  <td align="right">
<math>0</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(n - m + \tfrac12)  Q^m_{-1 / 2 +n+1}(x)
- 2nx Q^m_{-1 / 2 + n}(x)
+ (n+m-\tfrac12)Q^m_{-1 / 2 +n-1}(x)
</math>
  </td>
</tr>
</table>
  </td>
</tr>
</table>
</div>

==Oblate Spheroidal Coordinates==
===Setup===
Here we adopt the cartesian to spheroidal coordinate transformation,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>x</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>[(a_0^2+\xi_1^2)(1-\xi_2^2) ]^{1 / 2} \xi_3 \, ,</math>
  </td>
</tr>
<tr>
  <td align="right"><math>y</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>[(a_0^2+\xi_1^2)(1-\xi_2^2) ]^{1 / 2} (1 - \xi_3^2)^{1 / 2} \, ,</math>
  </td>
</tr>
<tr>
  <td align="right"><math>z</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\xi_1 \xi_2 \, .</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>], p. 662
  </td>
</tr>
</table>

If we adopt the notation replacements, <math>(\xi_1, \xi_2, \xi_3) \rightarrow (a_0 \xi, \eta, \cos\phi)</math>, we have,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>x</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>a_0 [(1+\xi^2)(1-\eta^2) ]^{1 / 2} \cos\phi \, ,</math>
  </td>
</tr>
<tr>
  <td align="right"><math>y</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>a_0 [(1+\xi^2)(1-\eta^2) ]^{1 / 2} \sin\phi \, ,</math>
  </td>
</tr>
<tr>
  <td align="right"><math>z</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>a_0 \xi \eta \, .</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>], p. 1292, Eq. (10.3.55)<br />
{{ Bardeen71 }}, &sect;IV, p. 428, Eq. (12)<br />
{{ HE84 }}, &sect;2, p. 498, Eq. (1)
  </td>
</tr>
</table>
The accompanying scale factors are,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>h_\xi</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>a_0 \biggl[ \frac{\xi^2 + \eta^2}{\xi^2 + 1} \biggr]^{1 / 2}
\, ,</math>
  </td>
</tr>
<tr>
  <td align="right"><math>h_\eta</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>a_0 \biggl[ \frac{\xi^2 + \eta^2}{1 - \eta^2} \biggr]^{1 / 2}
\, ,</math>
  </td>
</tr>
<tr>
  <td align="right"><math>h_\phi</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>a_0 (1 + \xi^2)^{ 1 / 2}(1 - \eta^2)^{1 / 2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>], p. 1292, Eq. (10.3.55)
  </td>
</tr>
</table>
in which case, the volume element is,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>d^3x</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
h_\xi h_\eta h_\phi \cdot  d\xi~ d\eta~ d\phi
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>a_0^3 
\biggl\{ 
\biggl[ \frac{\xi^2 + \eta^2}{\xi^2 + 1} \biggr]^{1 / 2}
\biggl[ \frac{\xi^2 + \eta^2}{1 - \eta^2} \biggr]^{1 / 2}
(1 + \xi^2)^{ 1 / 2}(1 - \eta^2)^{1 / 2}
\biggr\}d\xi~ d\eta~ d\phi 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
a_0^3 (\xi^2 + \eta^2) d\xi~ d\eta~ d\phi 
</math>
  </td>
</tr>
</table>

Now, according to [<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>], the Green's function in this oblate spheroidal coordinate system is given by the expression,

<table border="0" align="center">
<tr>
  <td align="right">
<math>\frac{1}{|\vec{x}^{~'} - \vec{x}|} = \frac{1}{R}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
~ \sum_{m=0}^{n}~ \epsilon_m i^{m+1}\biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
\cos[m(\phi - \phi')]P_n^m(\eta') \cdot
P_n^m(\eta)\biggl\{ \begin{array}{ll}
P_n^m(i\xi') Q_n^m(i\xi)\, ; ~~~ \xi > \xi' \\
P_n^m(i\xi) Q_n^m(i\xi')\, ; ~~~ \xi' > \xi \\
\end{array}
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>], Ch. 10, p. 1296, Eq. (10.3.63)
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
~ \sum_{m=0}^{n}~ \epsilon_m i^{m+1}\biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
\cos[m(\phi - \phi')]P_n^m(\eta') \cdot
P_n^m(\eta)\biggl\{ \begin{array}{ll}
(-i)^{-(n+m)} p_n^m(\xi') (-i)^{n -  2m + 1} q_n^m(\xi)\, ; ~~~ \xi > \xi' \\
(-i)^{-(n+m)} p_n^m(\xi) (-i)^{n -  2m + 1} q_n^m(\xi')\, ; ~~~ \xi' > \xi \\
\end{array}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
~ \sum_{m=0}^{n}~ \epsilon_m \biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
\cos[m(\phi - \phi')]P_n^m(\eta') \cdot
P_n^m(\eta)\biggl\{ \begin{array}{ll}
p_n^m(\xi') q_n^m(\xi)\, ; ~~~ \xi > \xi' \\
p_n^m(\xi) q_n^m(\xi')\, ; ~~~ \xi' > \xi \\
\end{array}
</math>
  </td>
</tr>
</table>
where,

<table border="0" align="center">
<tr>
  <td align="right">
<math>\epsilon_m</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \begin{array}{ll}
1~~~(m=0)\\
2~~~(m > 0)\\
\end{array}
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
{{ HE84 }} &sect;2, p. 498
  </td>
</tr>
</table>
and, following {{ HE84 }}, we have made the substitutions,

<table border="0" align="center">
<tr>
  <td align="right">
<math>P_n^m(i\xi)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(-i)^{-(n+m)} p_n^m(\xi) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>Q_n^m(i\xi)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(-i)^{n -  2m + 1} q_n^m(\xi) \, .</math>
  </td>
</tr>
</table>


<table border="1" align="center" width="90%" cellpadding="8"><tr><td align="left">
'''Oblate Spheroidal Coordinates:'''

According to the ''Erratum'' of &hellip;
<div align="center">{{ CTRS00figure }}</div>
the "expansion formula" that is relevant to oblate spheroidal coordinates is given correctly on p. 218 (Eq. 41) of MacRobert (1947).  It is,

<table border="0" align="center">
<tr>
  <td align="right">
<math>\frac{1}{|\vec{x} - \vec{x}^{~'}|} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{i}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
~ \sum_{m=0}^{n}~ \epsilon_m (-1)^{m}\biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
\cos[m(\phi - \phi')]P_n^m(\cos\theta') \cdot
P_n^m(\cos\theta)\biggl\{ \begin{array}{ll}
P_n^m(i\sinh\sigma') Q_n^m(i\sinh\sigma)\, ; ~~~ \sinh\sigma > \sinh\sigma' \\
P_n^m(i\sinh\sigma) Q_n^m(i\sinh\sigma')\, ; ~~~ \sinh\sigma' > \sinh\sigma \\
\end{array}
</math>
  </td>
</tr>

</table>

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

===Expression for the Potential due to General 3D Mass Distribution===
Hence, in spheroidal coordinates, the integral representation of the Poisson equation can be written as the sum of two terms &#8212; one in which the "radial" component of the volume integral covers the region, <math>0 < \xi' \le \xi</math>, while the other covers the region, <math>\xi < \xi' \le \infty</math> &#8212; namely, 

<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 \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^'</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-a_0^3 G \int_0^\xi d\xi' \int_{-1}^1 d\eta' \int_0^{2\pi} d\phi' \biggl\{ \frac{1}{R} \biggr\}_{\xi' < \xi} [(\xi')^2 + (\eta')^2] \rho(\xi', \eta', \phi') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-a_0^3 G \int_\xi^\infty d\xi' \int_{-1}^1 d\eta' \int_0^{2\pi} d\phi' \biggl\{ \frac{1}{R} \biggr\}_{\xi' > \xi} [(\xi')^2 + (\eta')^2] \rho(\xi', \eta', \phi') \, .
</math>
  </td>
</tr>
</table>
Inserting the appropriate expression for <math>1/R</math> in both terms gives,

<table border="0" align="center">
<tr>
  <td align="right">
<math>\Phi(\xi, \eta, \phi)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-a_0^3 G \int_0^\xi d\xi' \int_{-1}^1 d\eta' \int_0^{2\pi} d\phi' 
\biggl\{ 
\frac{2}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
~ \sum_{m=0}^{n}~ \epsilon_m \biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
\cos[m(\phi - \phi')]P_n^m(\eta') \cdot
P_n^m(\eta) \cdot p_n^m(\xi') q_n^m(\xi)
\biggr\} 
[(\xi')^2 + (\eta')^2] \rho(\xi', \eta', \phi') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-a_0^3 G \int_\xi^\infty d\xi' \int_{-1}^1 d\eta' \int_0^{2\pi} d\phi' 
\biggl\{ 
\frac{2}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
~ \sum_{m=0}^{n}~ \epsilon_m \biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
\cos[m(\phi - \phi')]P_n^m(\eta') \cdot
P_n^m(\eta) \cdot p_n^m(\xi) q_n^m(\xi')
\biggr\} 
[(\xi')^2 + (\eta')^2] \rho(\xi', \eta', \phi') 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-2a_0^2 G 
\sum_{n=0}^{\infty}~ (2n+1)
~ \sum_{m=0}^{n}~ \epsilon_m \biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
q_n^m(\xi)P_n^m(\eta) \cdot 
\int_0^\xi d\xi' \int_{-1}^1 d\eta' \int_0^{2\pi} d\phi' 
\biggl\{ 
\cos[m(\phi - \phi')]p_n^m(\xi')P_n^m(\eta')
\biggr\} 
[(\xi')^2 + (\eta')^2] \rho(\xi', \eta', \phi') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-2a_0^2 G 
~ \sum_{n=0}^{\infty}~ (2n+1)
~ \sum_{m=0}^{n}~ \epsilon_m \biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
p_n^m(\xi)P_n^m(\eta) \cdot  \int_\xi^\infty d\xi' \int_{-1}^1 d\eta' \int_0^{2\pi} d\phi' 
\biggl\{ 
\cos[m(\phi - \phi')]q_n^m(\xi')P_n^m(\eta') 
\biggr\} 
[(\xi')^2 + (\eta')^2] \rho(\xi', \eta', \phi') \, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
{{ HE84 }} &sect;2, p. 498, Eq. (2)
  </td>
</tr>
</table>
<font color="red">NOTE:</font>&nbsp; This expression for the potential exactly matches Eq. (2) (p. 498) of {{ HE84 }} '''except''' ours is a factor of two larger.

===Axisymmetric Mass Distribution===

For an axisymmetric mass distribution, we need only consider the <math>m = 0</math> contribution.  Also switching the integer index notation from <math>n</math> to <math>\ell</math> gives,

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

<tr>
  <td align="right">
<math>\Phi(\xi, \eta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-2a_0^2 G 
\sum_{\ell=0}^{\infty}~ (2\ell+1)
q_\ell(\xi)P_\ell(\eta) \cdot 
\int_0^\xi d\xi' \int_{-1}^1 d\eta'  
\biggl\{ 
p_\ell(\xi')P_\ell(\eta')
\biggr\} 
[(\xi')^2 + (\eta')^2] \rho(\xi', \eta') 
\int_0^{2\pi} d\phi'
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-2a_0^2 G 
~ \sum_{\ell=0}^{\infty}~ (2\ell+1)
p_\ell(\xi)P_\ell(\eta) \cdot  \int_\xi^\infty d\xi' \int_{-1}^1 d\eta'  
\biggl\{ 
q_\ell(\xi')P_\ell(\eta') 
\biggr\} 
[(\xi')^2 + (\eta')^2] \rho(\xi', \eta') 
\int_0^{2\pi} d\phi'
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 
\sum_{\ell=0}^{\infty}~ (2\ell+1)a_0^2
q_\ell(\xi)P_\ell(\eta) \cdot 
\int_0^\xi d\xi' \int_{-1}^1 d\eta'  
\biggl\{ 
p_\ell(\xi')P_\ell(\eta')
\biggr\} 
[(\xi')^2 + (\eta')^2] \biggl[4\pi G \rho(\xi', \eta') \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- 
~ \sum_{\ell=0}^{\infty}~ (2\ell+1)a_0^2
p_\ell(\xi)P_\ell(\eta) \cdot  \int_\xi^\infty d\xi' \int_{-1}^1 d\eta'  
\biggl\{ 
q_\ell(\xi')P_\ell(\eta') 
\biggr\} 
[(\xi')^2 + (\eta')^2]  
\biggl[4\pi G \rho(\xi', \eta') \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
{{ Bardeen71 }}, p. 429, Eq. (15)
  </td>
</tr>
</table>
which matches Eq. (15) of {{ Bardeen71 }} except, again, our expression gives a value for the potential that is a factor of two larger.  Finally, if we only consider even values of the index, <math>\ell</math> &#8212; in which case we make the replacement, <math>\ell \rightarrow 2n</math> &#8212; the expression for the potential becomes,
<table border="0" align="center">

<tr>
  <td align="right">
<math>\Phi(\xi, \eta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 
\sum_{n=0}^{\infty}~ (4n+1)a_0^2
q_{2n}(\xi)P_{2n}(\eta) \cdot 
\int_0^\xi d\xi' \int_{-1}^1 d\eta'  
\biggl\{ 
p_{2n}(\xi')P_{2n}(\eta')
\biggr\} 
[(\xi')^2 + (\eta')^2] \biggl[4\pi G \rho(\xi', \eta') \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- 
~ \sum_{\ell=0}^{\infty}~ (4n+1)a_0^2
p_{2n}(\xi)P_{2n}(\eta) \cdot  \int_\xi^\infty d\xi' \int_{-1}^1 d\eta'  
\biggl\{ 
q_{2n}(\xi')P_{2n}(\eta') 
\biggr\} 
[(\xi')^2 + (\eta')^2]  
\biggl[4\pi G \rho(\xi', \eta') \biggr] \, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
{{ HE83 }}, &sect;A.1, p. 587, Eq. (2)
  </td>
</tr>
</table>