Editing AxisymmetricConfigurations/PoissonEq (section)

==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>