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=Universit&eacute; de Bordeaux=

{{LSU_HBook_header}}

==Spheroid-Ring Systems==
Through a research collaboration at the [https://www.u-bordeaux.com Universit&eacute; de Bordeaux], [https://ui.adsabs.harvard.edu/abs/2019MNRAS.487.4504B/abstract B. Basillais &amp; J. -M. Hur&eacute; (2019), MNRAS, 487, 4504-4509] have published a paper titled, ''Rigidly Rotating, Incompressible Spheroid-Ring Systems:  New Bifurcations, Critical Rotations, and Degenerate States.''


==Exterior Gravitational Potential of Toroids==
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract J. -M. Hur&eacute;, B. Basillais, V. Karas, A. Trova, &amp; O. Semer&aacute;k (2020), MNRAS, 494, 5825-5838] have published a paper titled, ''The Exterior Gravitational Potential of Toroids.''  Here we examine how their work relates to the published work by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279)], which we have separately [[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential|discussed in detail]].

===Their Presentation===
On an initial reading, it appears as though the most relevant section of the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)] paper is &sect;8 titled, ''The Solid Torus.''  They write the gravitational potential in terms of the series expansion,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Psi_\mathrm{grav}(\vec{r})</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\Psi_0 + \sum\limits_{n=1}^N \Psi_n \, ,
</math>
  </td>
</tr>
</table>

[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;7, p. 5831, Eq. (42)
</div>
where, after setting <math>~M_\mathrm{tot} = 2\pi^2\rho_0 b^2 R_c </math> and acknowledging that <math>~V_{0,0} = 1 \, ,</math> we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Psi_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{GM_\mathrm{tot}}{r} \biggl[ \frac{r}{\Delta_0} \cdot \frac{2}{\pi} \boldsymbol{K}(k_0) \biggr]
</math>
  </td>
</tr>
</table>

[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;8, p. 5832, Eqs. (52) &amp; (53)
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{e^2} \biggl[ \Psi_1 + \Psi_2 \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{G\pi \rho_0 R_c b^2}{4 (k')^2 \Delta_0^3} \biggl\{
[\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k) - (k')^2 \Delta_0^2 \boldsymbol{K}(k)
\biggr\} \, .
</math>
  </td>
</tr>
</table>

[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;8, p. 5832, Eq. (54)
</div>
Note that the argument of the elliptic integral functions is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{2\sqrt{\varpi R}}{\Delta}
</math>
  </td>
<td align="center">&nbsp; &nbsp; where, &nbsp; &nbsp;</td>
  <td align="right">
<math>~\Delta</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ (R + \varpi)^2 + (Z-z)^2 \biggr]^{1 / 2} \, .
</math>
  </td>
</tr>
</table>

[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;2, p. 5826, Eqs. (4) &amp; (5)
</div>

===Our Presentation of Wong's (1973) Result===

====Setup====

From our [[User:Tohline/Apps/Wong1973Potential#D0andCn|accompanying discussion of Wong's (1973) derivation]], the exterior potential is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\sum_{n=0}^{\mathrm{nmax}} \epsilon_n \cos(n\theta) C_n(\cosh\eta_0)P_{n-\frac{1}{2}}(\cosh\eta) \, ,
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~D_0 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{2^{3/2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] 
=
\frac{2^{3/2} }{3\pi^2} \biggl[\frac{(R^2 - d^2)^{3 / 2}}{d^2 R}  \biggr] 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C_n(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(\cosh \eta_0) Q_{n - \frac{1}{2}}^2(\cosh \eta_0) 
- (n - \tfrac{3}{2}) Q_{n - \frac{1}{2}}(\cosh \eta_0)~Q^2_{n + \frac{1}{2}}(\cosh \eta_0) \, 
</math>
  </td>
</tr>
</table>
and where, in terms of the major ( R ) and minor ( d ) radii of the torus &#8212; or their ratio, &epsilon; &equiv; d/R, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cosh\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R}{d} = \frac{1}{\epsilon} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\sinh\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{d} = \frac{1}{d}\biggl[ R^2 - d^2 \biggr]^{1 / 2} = \frac{1}{\epsilon} \biggl[1 - \epsilon^2 \biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>

These expressions incorporate a number of [[User:Tohline/2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System|basic elements of a toroidal coordinate system]].  In what follows, we will also make use of the following relations:

<table border="1" width="80%" cellpadding="8" align="center"><tr><td align="left">
Once the primary scale factor, <math>~a</math>, has been specified, the illustration shown at the bottom of this inset box &#8212; see also our [[User:Tohline/Apps/DysonWongTori#Self-Gravitating.2C_Incompressible_.28Dyson-Wong.29_Tori|accompanying set of similar figures]] used by other researchers &#8212; helps in explaining how transformations can be made between any two of the referenced coordinate pairs:  <math>~(\varpi, z)</math>, <math>~(\eta, \theta)</math>, <math>~(r_1, r_2)</math>.

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

<tr>
  <td align="right">
<math>~\varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a\sinh\eta}{(\cosh\eta - \cos\theta)}</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;<math>~\Rightarrow ~</math>&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cosh\eta - \frac{a\sinh\eta}{\varpi}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z - Z_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a\sin\theta}{(\cosh\eta - \cos\theta)}</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;<math>~\Rightarrow ~</math>&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\sin\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(z - Z_0)}{\varpi} \cdot \sinh\eta </math>
  </td>
</tr>
</table>
Given that (sin<sup>2</sup>&theta; + cos<sup>2</sup>&theta;) = 1, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{(z - Z_0)}{\varpi} \cdot \sinh\eta \biggr]^2 + \biggl[\cosh\eta - \frac{a\sinh\eta}{\varpi}\biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \coth\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2a\varpi}\biggl[\varpi^2 + a^2 + (z - Z_0)^2  \biggr] \, .
</math>
  </td>
</tr>
</table>
We deduce as well that,

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

<tr>
  <td align="right">
<math>~\frac{2}{\coth\eta + 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \, ,
</math>&nbsp; &nbsp; &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~\sinh\eta + \cosh\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\varpi^2 + a^2 + (z - Z_0)^2}{(\varpi + a)^2 + (z - Z_0)^2} \, .
</math>
  </td>
</tr>
</table>

----
Given the definitions, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\varpi + a)^2 + (z - Z_0)^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\varpi - a)^2 + (z - Z_0)^2 \, ,</math>
  </td>
</tr>
</table>
we can use the transformations,

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

<tr>
  <td align="right">
<math>~\varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(r_1^2 - r_2^2)}{4a}</math> &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~(z - Z_0)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_2^2 - \frac{1}{16a^2}\biggl[ r_1^2 - r_2^2 - 4a^2 \biggr]^2 \, ,</math> &nbsp; &nbsp; or,
  </td>
</tr>

<tr>
  <td align="right">
<math>~(z - Z_0)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_1^2 - \frac{1}{16a^2}\biggl[ r_1^2 - r_2^2 + 4a^2 \biggr]^2 \, .</math>
  </td>
</tr>
</table>
Or we can use the transformations,

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

<tr>
  <td align="right">
<math>~\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ln \biggl(\frac{r_1}{r_2}\biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \, .</math>
  </td>
</tr>
</table>

----

Additional potentially useful relations can be found in an [[User:Tohline/2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|accompanying chapter wherein we present a variety of basic elements of a toroidal coordinate system]].

[[File:WongTorusIllustration02.png|400px|center|Wong diagram]]
</td></tr></table>

====Leading (n  = 0) Term====
=====Wong's Expression=====
Now, from our [[User:Tohline/Apps/Wong1973Potential#Attempt_.232|separate derivation]] we have,

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

<tr>
  <td align="right">
<math>~P_{-1 / 2}(\cosh\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} Q_{-1 / 2}(\coth\eta) \, .
</math>
  </td>
</tr>
</table>
<span id="KeyEquation">And if we make the function-argument substitution,</span> <math>~z \rightarrow \coth\eta</math>, in the "[[User:Tohline/Appendix/Equation_templates#Analytic_Expressions_.26_Plots|Key Equation]],"

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

<tr>
<td align="right">
[[Image:LSU_Key.png|25px|link=http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#Toroidal_Function_Evaluations]]
</td> 
  <td align="right">
<math>~Q_{-\frac{1}{2}}(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="center" colspan="4">
[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 Abramowitz &amp; Stegun (1995)], p. 337, eq. (8.13.3)
  </td>
</table>
we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_{-1 / 2}(\cosh\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k) \, ,
</math>
  </td>
</tr>
</table>

where, from above, we recognize that,
<div align="center">
<math>~
k \equiv \biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2}  
=
\biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} \, .
</math>
</div>

So, the leading (n = 0) term gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~C_0(\cosh\eta_0)P_{-\frac{1}{2}}(\cosh\eta) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0~C_0(\cosh\eta_0)
\biggl[ \frac{a \sinh\eta}{\varpi} \biggr]^{1 / 2} ~\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{D_0~C_0(\cosh\eta_0)}{\pi}
\biggl[ \frac{2a }{\varpi} \biggr]^{1 / 2} ~ k \boldsymbol{K}(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- C_0(\cosh\eta_0) \cdot \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] 
\frac{a}{ [ (\varpi + a)^2 + (z - Z_0)^2 ]^{1 / 2} } \cdot  \boldsymbol{K}(k) \, .
</math>
  </td>
</tr>
</table>

=====Thin-Ring Evaluation of C<sub>0</sub>=====
In an [[User:Tohline/Apps/Wong1973Potential#Thin_Ring_Approximation|accompanying discussion of the thin-ring approximation]], we showed that as <math>~\cosh\eta_0 \rightarrow \infty</math>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~C_0(x)\biggr|_{x\rightarrow \infty}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{3 \pi^2}{2^2} \biggr) \frac{1}{\cosh^2\eta_0} \, .
</math>
  </td>
</tr>
</table>
Hence, in this limit we can write,

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

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\eta,\theta)\biggr|_\mathrm{thin-ring}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{2 }{\pi}  \cancelto{1}{\biggl[\frac{\sinh\eta_0}{\cosh\eta_0}\biggr]^3 } 
\frac{a}{ [ (\varpi + a)^2 + (z - Z_0)^2 ]^{1 / 2} } \cdot  \boldsymbol{K}(k) \, .
</math>
  </td>
</tr>
</table>

=====More General Evaluation of C<sub>0</sub>=====

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
<font color="red">NOTE of CAUTION:</font>  In our [[#KeyEquation|above evaluation of the toroidal function]], <math>~Q_{-\frac{1}{2}}(z)</math>, we appropriately associated the function argument, <math>~z</math>, with the hyperbolic-cotangent of <math>~\eta</math>; that is, we made the substitution, <math>~z \rightarrow \coth\eta</math>.  Here, as we assess the behavior of, and evaluate, the leading coefficient, <math>~C_0</math>, an alternate substitution is appropriate, namely, <math>~z_0 \rightarrow \cosh\eta_0</math>; we affix the subscript zero to this function argument in an effort to minimize possible confusion with the argument, <math>~z</math>.
</td></tr></table>

Drawing from our [[User:Tohline/Appendix/Equation_templates#Analytic_Expressions_.26_Plots|accompanying tabulation of ''Toroidal Function Evaluations'']], we have more generally,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2C_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ Q_{+\frac{1}{2}}(\cosh \eta_0) \biggr]
\biggl[ Q_{ - \frac{1}{2}}^2(\cosh \eta_0) \biggr]
+ 
3 \biggl[ Q_{ - \frac{1}{2}}(\cosh \eta_0) \biggr]
\biggl[ Q^2_{ + \frac{1}{2}}(\cosh \eta_0) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \cosh\eta_0 ~k_0~K(k_0) ~-~ [2(\cosh\eta_0+1)]^{1 / 2} E(k_0) \biggr]
\times \biggl\{ \frac{ 4 \cosh\eta_0 ~E(k_0) - (\cosh\eta_0-1) K(k_0) }{ [2^{3} (\cosh\eta_0+1) (\cosh\eta_0-1)^{2} ]^{1 / 2}} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 
\frac{3}{2^2} \biggl[ k_0 ~K ( k_0) \biggr]
\times \biggl\{ \cosh\eta_0~ k_0~K ( k_0 )  
~-~(\cosh^2\eta_0+3) \biggl[ \frac{2}{(\cosh\eta_0 - 1)(\cosh^2\eta_0 -1)} \biggr]^{1 / 2} E(k_0)
 \biggr\} \, ,
</math>
  </td>
</tr>
</table>
<span id="FirstEvaluations">where,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k_0</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2}{\cosh\eta_0+1}\biggr]^{1 / 2} ~~~\Rightarrow ~~~ (\cosh\eta_0 + 1) = \frac{2}{k_0^2} \, .</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="10">
<tr><td align="left">
Looking back at our [[User:Tohline/Apps/Wong1973Potential#Exterior_Solution_.28n_.3D_0.29|previous numerical evaluation]] of <math>~C_0(\cosh\eta_0)</math> when <math>~z_0 = \cosh\eta_0 = 3 ~~\Rightarrow ~~~ k_0 = 2^{-1 / 2}</math>, we see that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
[[User:Tohline/Appendix/Equation_templates#Toroidal_Function_Evaluations|Appendix Expression:]] <math>~Q_{-\tfrac{1}{2}}(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~k_0 K(k_0)</math>
  </td>
</tr>

<tr>
  <td align="right">
Hence [[User:Tohline/Appendix/Equation_templates#Comparison_with_Table_IX_from_MF53|MF53 value]], <math>~Q_{-\tfrac{1}{2}}(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1.311028777 ~~~\Rightarrow ~~~ K(k_0) = 1.854074677</math>
  </td>
</tr>

<tr>
  <td align="right">
[[User:Tohline/Appendix/Equation_templates#Toroidal_Function_Evaluations|Appendix Expression:]] <math>~Q_{+\tfrac{1}{2}}(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z~k_0 K(k_0) - [2(z+1)]^{1 / 2} E(k_0)</math>
  </td>
</tr>

<tr>
  <td align="right">
Hence [[User:Tohline/Appendix/Equation_templates#Comparison_with_Table_IX_from_MF53|MF53 value]], <math>~Q_{+\tfrac{1}{2}}(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.1128885424 ~~~\Rightarrow~~~ E(k_0) = 1.350643881</math>
  </td>
</tr>

<tr>
  <td align="right">
[[User:Tohline/Appendix/Equation_templates#Toroidal_Function_Evaluations|Appendix Expression:]] <math>~Q^2_{-\tfrac{1}{2}}(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[2^3(z-1)(z^2-1)]^{-1 / 2} [4zE(k_0) - (z-1)K(k_0)]</math>
  </td>
</tr>

<tr>
  <td align="right">
Hence, <math>~Q^2_{-\tfrac{1}{2}}(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1.104816977</math>, which matches  [[User:Tohline/Appendix/Equation_templates#Comparison_with_Table_IX_from_MF53|MF53 value]]
  </td>
</tr>

<tr>
  <td align="right">
[[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Evaluating_Q2.CE.BD|Additional derivation:]] <math>~Q^2_{+\tfrac{1}{2}}(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{1}{2^2}\biggl\{ z k_0~K ( k_0 )  
~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
Hence, <math>~Q^2_{+\tfrac{1}{2}}(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.449302588</math>
  </td>
</tr>
</table>

----

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ C_0(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2}~Q_{+\frac{1}{2}}(3) \cdot Q_{- \frac{1}{2}}^2(3) 
+ \frac{3}{2}~ Q_{- \frac{1}{2}}(3)\cdot Q^2_{+ \frac{1}{2}}(3) 
=
0.945933522 \, .
</math>
  </td>
</tr>
</table>

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


Attempting to simplify this expression, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2C_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \cosh\eta_0 ~k_0~K(k_0) ~-~ \biggl(\frac{2}{k_0}\biggr) E(k_0) \biggr\}
\times \biggl\{ \frac{ 4 \cosh\eta_0 ~E(k_0) - (\cosh\eta_0-1) K(k_0) }{ [2^{2} k_0^{-1} (\cosh\eta_0-1) ]} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 
\frac{3}{2^2} \biggl[ k_0 ~K ( k_0) \biggr]
\times \biggl\{ \cosh\eta_0~ k_0~K ( k_0 )  
~-~(\cosh^2\eta_0+3) \biggl[ \frac{k_0^2}{(\cosh\eta_0 - 1)^2} \biggr]^{1 / 2} E(k_0)
 \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 2^3(\cosh\eta_0 - 1)C_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \cosh\eta_0 ~k_0^2~K(k_0) ~-~ 2 E(k_0) \biggr\}
\times \biggl\{ 4 \cosh\eta_0 ~E(k_0) - (\cosh\eta_0-1) K(k_0)  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 
3 k_0 ~K ( k_0) 
\times \biggl\{ \cosh\eta_0(\cosh\eta_0 - 1)~ k_0~K ( k_0 )  
~-~(\cosh^2\eta_0+3) k_0 E(k_0)
 \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~K(k_0)\cdot K(k_0) \biggl[ (\cosh\eta_0-1) \cdot \cosh\eta_0 ~k_0^2 + 3\cosh\eta_0~ (\cosh\eta_0~-1)k_0^2\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ K(k_0)\cdot E(k_0) \biggl[ 2^2 \cosh^2\eta_0 ~k_0^2  + 2(\cosh\eta_0 ~-1)  + 3k_0^2 (\cosh^2\eta_0 ~ + 3)\biggr] 
- E(k_0)\cdot E(k_0) \biggl[2^3\cosh\eta_0  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{ 2^3(\cosh\eta_0 - 1)}{k_0^2} \biggr] C_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~K(k_0)\cdot K(k_0) \biggl[ (\cosh\eta_0-1) \cdot \cosh\eta_0  + 3\cosh\eta_0~ (\cosh\eta_0~-1) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ K(k_0)\cdot E(k_0) \biggl[ 2^2 \cosh^2\eta_0   + \frac{2}{k_0^2}(\cosh\eta_0 ~-1)  + 3 (\cosh^2\eta_0 ~ + 3)\biggr] 
- E(k_0)\cdot E(k_0) \biggl[\frac{2^3\cosh\eta_0}{k_0^2}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (\cosh^2\eta_0 - 1) C_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
K(k_0)\cdot K(k_0) \biggl[ \cosh\eta_0(1 - \cosh\eta_0)  \biggr] 
+ 2K(k_0)\cdot E(k_0) \biggl[ \cosh^2\eta_0  + 1\biggr] 
- E(k_0)\cdot E(k_0) \biggl[ \cosh\eta_0(1 + \cosh\eta_0)  \biggr]
</math>
  </td>
</tr>
</table>

This last, simplifed expression gives, as above, <math>~C_0(3) = 0.945933523</math>.  <font color="red">TERRIFIC!</font>

Finally then, for any choice of <math>~\eta_0</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\eta,\theta)\biggr|_\mathrm{exterior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{2^{3} }{3\pi^3} 
\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] 
\frac{a}{ [ (\varpi + a)^2 + (z - Z_0)^2 ]^{1 / 2} } \cdot  \boldsymbol{K}(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \biggl\{
K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ] 
+ 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0  + 1 ] 
- E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0)  ]
\biggr\}  \, .
</math>
  </td>
</tr>
</table>

====Second (n  = 1) Term====
The second (n = 1) term in [[User:Tohline/Apps/Wong1973Potential#D0andCn|Wong's (1973) expression for the exterior potential]] is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0
(\cosh\eta - \cos\theta)^{1 / 2} \cdot 2 \cos\theta \cdot  C_1(\cosh\eta_0)P_{+\frac{1}{2}}(\cosh\eta) \, ,
</math>
  </td>
</tr>
</table>
where, <math>~D_0</math> is the same as [[#Setup|above]], and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~C_1(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\tfrac{3}{2} Q_{+\frac{3}{2}}(\cosh \eta_0) Q_{+\frac{1}{2}}^2(\cosh \eta_0) 
+ \tfrac{1}{2} Q_{+\frac{1}{2}}(\cosh \eta_0)~Q^2_{+ \frac{3}{2}}(\cosh \eta_0) \, .
</math>
  </td>
</tr>
</table>
Now, from our [[User:Tohline/Appendix/Equation_templates#Toroidal_Function_Evaluations|accompanying table of "Toroidal Function Evaluations"]], it appears as though,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_{+\frac{1}{2}}(\cosh\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sqrt{2}}{\pi} (\sinh\eta)^{+1 / 2} k^{-1} E(k) \, ,</math>
  </td>
</tr>
</table>

where, as above,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2}{\coth\eta+1} \biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>
Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] C_1(\cosh\eta_0) 
\biggl[ \cos\theta \cdot (\cosh\eta - \cos\theta)^{1 / 2}  (\sinh\eta)^{+1 / 2} \biggr] 
k^{-1} E(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] C_1(\cosh\eta_0) \cdot \cos\theta
\biggl\{ \frac{a\sinh^2\eta}{\varpi}  \cdot \frac{\coth\eta + 1}{2} \biggr\}^{1 / 2} 
E(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] C_1(\cosh\eta_0) \cdot \cos\theta
\biggl\{ \biggl( \frac{a}{2\varpi} \biggr) \biggl[ \frac{r_1^2 - r_2^2}{2r_1 r_2} \biggr]^2 \cdot \biggl[ \frac{2r_1^2}{r_1^2 - r_2^2} \biggr] \biggr\}^{1 / 2} 
E(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] C_1(\cosh\eta_0) \cdot \cos\theta
\biggl\{ \biggl( \frac{a}{2} \biggr)\biggl[ \frac{4a}{r_1^2 - r_2^2} \biggr] \biggl[ \frac{r_1^2 - r_2^2}{2r_1 r_2} \biggr]^2 \cdot \biggl[ \frac{2r_1^2}{r_1^2 - r_2^2} \biggr] \biggr\}^{1 / 2} 
E(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{2^{3} a}{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] C_1(\cosh\eta_0) 
\biggl[ \frac{\cos\theta}{r_2} \biggr] 
E(k)
= 
- \frac{2^{3} a}{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] C_1(\cosh\eta_0) 
\biggl[ \frac{\cos\theta}{\sqrt{ (\varpi - a)^2 + (z-Z_0)^2 }} \biggr] 
E(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{2^{3} a}{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr]  C_1(\cosh\eta_0) \biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2^2} \biggr]
\boldsymbol{E}(k) \, .
</math>
  </td>
</tr>
</table>


<table border="1" align="center" width="80%" cellpadding="10">
<tr><td align="left">
From the [[#FirstEvaluations|above function tabulations &amp; evaluations]] &#8212; for example, <math>~ K(k_0) = 1.854074677</math> and <math>~ E(k_0) = 1.350643881</math> &#8212; and a [[User:Tohline/Appendix/Equation_templates#Example_Recurrence_Relations|separate listing of ''Example Recurrence Relations'']], we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
[[User:Tohline/Appendix/Equation_templates#Toroidal_Function_Evaluations|Appendix Expression:]] <math>~Q_{-\tfrac{1}{2}}(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~k_0 K(k_0)</math>
  </td>
</tr>

<tr>
  <td align="right">
[[User:Tohline/Appendix/Equation_templates#Toroidal_Function_Evaluations|Appendix Expression:]] <math>~Q_{+\tfrac{1}{2}}(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z~k_0 K(k_0) - [2(z+1)]^{1 / 2} E(k_0)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~Q^0_{m-\tfrac{1}{2}}</math> [[User:Tohline/Appendix/Equation_templates#Example_Recurrence_Relations|recurrence]] with m = 2:  
<math>~Q_{+\tfrac{3}{2}}(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4}{3} z~Q_{+\tfrac{1}{2}}(z_0) - \frac{1}{3} Q_{-\tfrac{1}{2}}(z_0)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4}{3} z \{z~k_0 K(k_0) - [2(z+1)]^{1 / 2} E(k_0)\} - \frac{1}{3}k_0 K(k_0)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{3} \biggl[ (4z^2 - 1 )k_0 K(k_0) - 4 z[2(z+1)]^{1 / 2} E(k_0) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
Hence, <math>~Q_{+\tfrac{3}{2}}(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.014544576 \, .</math>
  </td>
</tr>
</table>

----

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
[[User:Tohline/Appendix/Equation_templates#Toroidal_Function_Evaluations|Appendix Expression:]] <math>~Q^2_{-\tfrac{1}{2}}(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[2^3(z-1)(z^2-1)]^{-1 / 2} [4zE(k_0) - (z-1)K(k_0)]</math>
  </td>
</tr>

<tr>
  <td align="right">
[[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Evaluating_Q2.CE.BD|Additional derivation:]] <math>~Q^2_{+\tfrac{1}{2}}(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{1}{2^2}\biggl\{ z k_0~K ( k_0 )  
~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\} 
</math>
  </td>
</tr>
</table>

Then, letting <math>~\mu \rightarrow 2</math> and, for all m &ge; 2, letting <math>~\nu \rightarrow (m - \tfrac{1}{2})</math> in the "Key Equation," 

{{ User:Tohline/Math/EQ_Toroidal04 }}

we have,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~(m - \tfrac{3}{2})Q^{2}_{m+\tfrac{1}{2}} (z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(2m)z Q_{m-\tfrac{1}{2}}^{2}(z) - (m + \tfrac{3}{2})Q^{2}_{m - \tfrac{3}{2}}(z) \, .
</math>
  </td>
</tr>
</table>

Therefore, specifically for m = 1, we obtain the recurrence relation,

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

<tr>
  <td align="right">
<math>~Q^{2}_{+\tfrac{3}{2}} (z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
5 Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0)  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
Hence, <math>~Q^{2}_{+\tfrac{3}{2}} (3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.132453829 \, . 
</math>
  </td>
</tr>
</table>

----

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ C_1(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{3}{2}~Q_{+\frac{3}{2}}(3) \cdot Q_{+ \frac{1}{2}}^2(3) 
+ \frac{1}{2}~ Q_{+ \frac{1}{2}}(3)\cdot Q^2_{+ \frac{3}{2}}(3) 
= 0.017278633 \, .
</math>
  </td>
</tr>
</table>

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


While keeping in mind that, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cosh\eta_0 \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~k_0^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{\cosh\eta_0 + 1} 
=
\frac{2}{z_0 + 1}
\, ,</math>
  </td>
</tr>
</table>

let's attempt to express this leading coefficient, <math>~C_1(\cosh\eta_0)</math>, entirely in terms of the pair of complete elliptic integral functions.
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2C_1(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3 \biggl[ Q_{+\frac{3}{2}}(z_0) \biggr]\times \biggl[ Q_{+\frac{1}{2}}^2(z_0) \biggr]
+ \biggl[ Q_{+\frac{1}{2}}(z_0) \biggr]\times \biggl[ 5 Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[3 Q_{+\frac{3}{2}}(z_0)  -4z Q_{+\frac{1}{2}}(z_0) \biggr]\times \biggl[ Q_{+\frac{1}{2}}^2(z_0) \biggr]
+ \biggl[ 5Q_{+\frac{1}{2}}(z_0) \biggr]\times \biggl[ Q^{2}_{- \tfrac{1}{2}}(z_0) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\frac{1}{2^2}\biggl\{  (4z^2 - 1 )k_0 K(k_0) - 4 z[2(z+1)]^{1 / 2} E(k_0)    
-4z \biggl[ z~k_0 K(k_0) - [2(z+1)]^{1 / 2} E(k_0) \biggr] \biggr\}
\times 
\biggl\{ z k_0~K ( k_0 )  
~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 5\biggl[ ~z~k_0 K(k_0) - [2(z+1)]^{1 / 2} E(k_0) \biggr]
\times 
\biggl\{ [2^3(z-1)(z^2-1)]^{-1 / 2} [4zE(k_0) - (z-1)K(k_0)] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^2} \cdot k_0 K(k_0) 
\times 
\biggl\{ z k_0~K ( k_0 )  
~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 5[2^3(z-1)(z^2-1)]^{-1 / 2} \biggl[ ~z~k_0 K(k_0) - [2(z+1)]^{1 / 2} E(k_0) \biggr]
\times 
\biggl\{ 4zE(k_0) - (z-1)K(k_0) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
K(k_0)\cdot K(k_0) \biggl\{ \frac{z k_0^2}{2^2} - 5[2^3(z-1)(z^2-1)]^{-1 / 2} \cdot zk_0(z-1)\biggr\}
+
E(k_0)\cdot E(k_0) \biggl\{ -5[2^3(z-1)(z^2-1)]^{-1 / 2}\cdot [2(z+1)]^{1 / 2} \cdot 4z  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~K(k_0)\cdot E(k_0) \biggl\{ 
-~\frac{1}{2^2} \cdot k_0(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2}  
+ 5[2^3(z-1)(z^2-1)]^{-1 / 2} \cdot 4z^2k_0
+ 5[2^3(z-1)(z^2-1)]^{-1 / 2}\cdot [2(z+1)]^{1 / 2} \cdot (z-1)
\biggr\} \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~2[(z-1)(z^2-1)]^{1 / 2} C_1(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
z k_0 \cdot K(k_0)\cdot K(k_0) \biggl\{ \frac{k_0}{2^2} \biggl[(z-1)(z^2-1) \biggr]^{1 / 2}  - \frac{5(z-1)}{2^{3/2}} \biggr\}
-~10 z(z+1)^{1 / 2}  \cdot E(k_0)\cdot E(k_0) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~2^{-3/2} K(k_0)\cdot E(k_0) \biggl\{ 
k_0[19z^2 - 3 ]   
+ 5(z-1) [2(z+1)]^{1 / 2}  
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~2^{3/2}\biggl[ \frac{(z-1)}{k_0} \biggr] C_1(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
z k_0 \cdot K(k_0)\cdot K(k_0) \biggl\{ \frac{k_0}{2^2} \biggl[\frac{2^{1 / 2}(z-1)}{k_0} \biggr]  - \frac{5(z-1)}{2^{3/2}} \biggr\}
-~\biggl[ \frac{2^{3 / 2} \cdot 5z}{k_0}  \biggr] E(k_0)\cdot E(k_0) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~2^{-3/2} K(k_0)\cdot E(k_0) \biggl\{ 
k_0[19z^2 - 3 ]   
+ \frac{10 (z-1)}{k_0}   
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~C_1(z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{2(3z^2 - 1)}{(z^2-1)}      \biggr]K(k_0)\cdot E(k_0) 
-~\biggl[ \frac{z}{(z+1)} \biggr] K(k_0)\cdot K(k_0) 
-~\biggl[ \frac{ 5z}{(z-1)}  \biggr] E(k_0)\cdot E(k_0) \, .
</math>
  </td>
</tr>
</table>


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