Editing 2DStructure/ToroidalCoordinates (section)

==Toroidal Coordinates==

[[2DStructure/ToroidalCoordinateIntegrationLimits#Toroidal-Coordinate_Integration_Limits|Click here]] to find a supporting, detailed illustration of Toroidal Coordinates and relevant integration limits.

===Properties===
Here we highlight certain properties and features of the [[Appendix/References#MF53|MF53]] toroidal coordinate system; more details can be found in a [[Appendix/Ramblings/ToroidalCoordinates#Toroidal_Coordinates|related set of our online notes]].  Most importantly in the context of our discussion, if (at all azimuthal angles) the origin of the toroidal coordinate system is placed at the ''cylindrical-coordinate'' location, <math>~(a, Z_0), </math> the pair of orthogonal coordinates, <math>~(\xi_1, \xi_2)</math>, is related to the cylindrical coordinate pair, <math>~(\varpi, Z)</math>, via the expressions,
<table align="center" border="0" cellpadding="5">
<tr><th align="center" colspan="1">&nbsp;</th>
       <th align="center" colspan="2">[<font color="red"><b>[[Appendix/References#MF5|MF53]]</b></font>]</th>
       <th align="center" colspan="2">[https://en.wikipedia.org/wiki/Toroidal_coordinates Wikipedia]</th>
</tr>
<tr>
  <td align="right">
<math>
~\frac{\varpi}{a} 
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} 
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{\sinh\tau}{\cosh\tau - \cos\theta} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{(Z_0 - Z)}{a}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\pm~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2} 
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{\sin\theta}{\cosh\tau - \cos\theta}  \, .
</math>
  </td>
</tr>
</table>

An off-center circle &#8212; such as the white circle with purple perimeter depicted in our Figure 1 diagram &#8212; is generated if a value of the "radial" coordinate, <math>~\xi_1</math>, is chosen from within the range, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
       <th align="center" colspan="1">[<font color="red"><b>[[Appendix/References#MF5|MF53]]</b></font>]</th>
       <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
       <th align="center" colspan="1">[https://en.wikipedia.org/wiki/Toroidal_coordinates Wikipedia]</th>
</tr>
<tr>
  <td align="right">
<math>~+1 \leq \xi_1 < \infty</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; or, equivalently &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="left">
<math>~0 \leq \tau < \infty \, ,</math>
  </td>
</tr>
</table>
</div>
and held fixed while the "angular" coordinate, <math>~\xi_2</math>, is varied over the range, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
       <th align="center" colspan="1">[<font color="red"><b>[[Appendix/References#MF5|MF53]]</b></font>]</th>
       <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
       <th align="center" colspan="1">[https://en.wikipedia.org/wiki/Toroidal_coordinates Wikipedia]</th>
</tr>
<tr>
  <td align="right">
<math>~ -1 \leq \xi_2 \leq +1</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; or, more completely &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="left">
<math>~-\pi < \theta \leq \pi</math>
  </td>
</tr>
</table>
</div>
Hereafter, we will refer to this <math>~\xi_1</math> = constant circle as a "<math>\xi_1</math>-circle."  A <math>\xi_1</math>-circle of radius zero and, hence, the origin of the toroidal coordinate system is associated with the ''upper'' limiting value of the radial coordinate, namely, <math>~\xi_1 = \infty</math>; as the value of <math>~\xi_1</math> is decreased monotonically, the radius of the circle (for example, the circle of radius, <math>~r_0</math>, in our Figure 1) steadily grows; and the radius of this circle becomes infinite at the radial coordinate's other limiting value, <math>~\xi_1 = 1</math>.

In the <math>~Z = Z_0</math> plane, the location of the inner and outer edges of the toroidal-coordinate surface are determined by setting <math>~\xi_2 = -1</math> (inner) and <math>~\xi_2 = +1</math> (outer).  Hence,
<table align="center" border="0" cellpadding="4">
<tr>
  <td align="right">
<math>
~\biggl(\frac{\varpi}{a}\biggr)_\mathrm{inner}
</math>
  </td>
  <td align="center>
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 +1} = \biggl[\frac{(\xi_1 - 1)}{(\xi_1 + 1)} \biggr]^{1/2} \, ,
</math>

  </td>
</tr>

<tr>
  <td align="right">
<math>
~\biggl(\frac{\varpi}{a}\biggr)_\mathrm{outer}
</math>
  </td>
  <td align="center>
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - 1} = \biggl[\frac{(\xi_1 + 1)}{(\xi_1 - 1)} \biggr]^{1/2} \, .
</math>

  </td>
</tr>
</table>
Hence, also, the (cylindrical) radial location of the "center" of each toroidal-coordinate surface &#8212; labeled <math>~R_0</math> in our Figure 1 &#8212; is given by the expression,
<div align="center">
<math>
R_0 = \frac{a}{2} \biggl[ \biggl(\frac{\varpi}{a}\biggr)_\mathrm{outer} 
+ \biggl(\frac{\varpi}{a}\biggr)_\mathrm{inner} \biggr] = \frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}}  \, ,
</math>
</div>
and the surface's cross-sectional radius &#8212; labeled <math>~r_0</math> in our Figure 1 &#8212; is given by the expression,
<div align="center">
<math>
r_0 = \frac{a}{2} \biggl[ \biggl(\frac{\varpi}{a}\biggr)_\mathrm{outer} 
- \biggl(\frac{\varpi}{a}\biggr)_\mathrm{inner} \biggr] = \frac{a}{(\xi_1^2 - 1)^{1/2}} \, .
</math>
</div>

This last expression quantifies, and its simplicity reinforces, our earlier statement; that is, as the value of <math>~\xi_1</math> is decreased monotonically, the radius of the circle, <math>~r_0</math>, steadily grows.  The next-to-last expression makes it clear, as well, that <math>~R_0</math> grows larger and, therefore, the location of the center of a <math>\xi_1</math>-circle shifts farther away from the symmetry axis as the value of <math>~\xi_1</math> is decreased.  Notice that, for any off-center circle, the ratio of these to lengths gives the value of the toroidal-coordinate system's dimensionless "radial" coordinate, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{R_0}{r_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}}\biggr] \biggl[ \frac{(\xi_1^2 - 1)^{1/2}}{a}\biggr]
= \xi_1 \, .</math>
  </td>
</tr>
</table>
</div>
Notice, furthermore, that there is a particular combination of these two lengths that is independent of <math>~\xi_1</math>, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_0 \biggl[\biggl( \frac{R_0}{r_0} \biggr)^2 - 1  \biggr]^{1/2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{(\xi_1^2 - 1)^{1/2}} \biggl[\xi_1^2 - 1  \biggr]^{1/2} = a \, .</math>
  </td>
</tr>
</table>
</div>
This is a manner in which one can determine the radial position, <math>~a</math>, of the origin of the toroidal coordinate system that could legitimately be associated with any particular off-center circle, such as the white circle with a purple perimeter drawn in our Figure 1.

===Connection With the Physical Problem===
[[#OffCenterCircle|Earlier]], we stated that the off-center circle with purple perimeter displayed in Figure 1 is prescribed by the algebraic expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(R_0 - \varpi)^2 + (Z_0 - Z)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_0^2 \, .</math>
  </td>
</tr>
</table>
</div>
Let's plug in the "toroidal-coordinate" expressions for each parameter that appears on the left-hand side of this relation and see whether, after simplification, it reduces to the right-hand side. 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
LHS
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(R_0 - \varpi)^2 + (Z_0 - Z)^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}}  - \frac{a(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \biggr]^2 
+ \biggl[\pm~\frac{a(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2}\biggr]^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a^2}{(\xi_1^2 - 1)} \biggl\{
\biggl[ \xi_1  - \frac{(\xi_1^2 - 1)}{\xi_1 - \xi_2} \biggr]^2 
+ \biggl[\frac{(1-\xi_2^2)^{1/2}(\xi_1^2 - 1)^{1/2} }{\xi_1 - \xi_2}\biggr]^2
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a^2}{(\xi_1^2 - 1)(\xi_1-\xi_2)^2} \biggl\{
\biggl[ \xi_1(\xi_1-\xi_2)  - (\xi_1^2 - 1) \biggr]^2 
+ (1-\xi_2^2)(\xi_1^2 - 1)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a^2}{(\xi_1^2 - 1)(\xi_1-\xi_2)^2} \biggl[
( 1-\xi_1\xi_2 )^2 + (\xi_1^2 - 1 -\xi_1^2\xi_2^2 + \xi_2^2)
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a^2}{(\xi_1^2 - 1)(\xi_1-\xi_2)^2} \biggl[
\xi_1^2  -2\xi_1\xi_2 +  \xi_2^2
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a^2}{(\xi_1^2 - 1)} \, .
</math>
  </td>
</tr>
</table>
</div>
This, indeed, equals the right-hand side of the relation, which is, <math>~r_0^2</math>.  It all nicely checks out!

Next, taking a hint from the EUREKA! moment recorded in [[Appendix/Ramblings/ToroidalCoordinates#Relating_CCGF_Expansion_to_Toroidal_Coordinates|our accompanying notes]], let's rewrite the function <math>~\Chi</math> in terms of toroidal rather than cylindrical coordinates, where <math>~\Chi</math> is the argument of the special function, <math>~Q_{-1/2}</math>, that appears in the [[2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential|above definition of <math>~q_0</math>]].  More specifically, let's assume that the coordinate location at which the gravitational potential is to be evaluated, <math>~(R_*, Z_*)</math>, is taken to be the cylindrical-coordinate location of the origin of the toroidal coordinate system, <math>~(a, Z_0)</math>.  Given this association, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Chi </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{R_*^2 + \varpi^2 + (Z_* - Z)^2}{2R_* \varpi}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a^2 + \varpi^2 + (Z_0 - Z)^2}{2a \varpi}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1 + (\varpi/a)^2 + [(Z_0 - Z)/a]^2}{2(\varpi/a) }</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\Chi^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 2 \biggl( \frac{\varpi}{a} \biggr) \biggr]^{-2}
\biggl\{ 1 + \biggl(\frac{\varpi}{a} \biggr)^2 + \biggl[\frac{(Z_0 - Z)}{a} \biggr]^2 \biggr\}^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \biggr]^{-2}
\biggl\{ 1 + \biggl[\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \biggr]^2 
+ \biggl[\pm~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2}  \biggr]^2 \biggr\}^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\xi_1 - \xi_2)^2}{4(\xi_1^2 - 1)} 
\biggl[ 1 + \frac{(\xi_1^2 - 1)}{(\xi_1 - \xi_2)^2} 
+ \frac{(1-\xi_2^2)}{(\xi_1 - \xi_2)^2}  \biggr]^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4(\xi_1^2 - 1)(\xi_1 - \xi_2)^2} 
\biggl[ (\xi_1 - \xi_2)^2 + (\xi_1^2 - 1)
+ (1-\xi_2^2)  \biggr]^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[ 2\xi_1(\xi_1 - \xi_2 ) ]^2}{4(\xi_1^2 - 1)(\xi_1 - \xi_2)^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\xi_1^2}{\xi_1^2 - 1} \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, when the function, <math>~q_0</math>, is rewritten in terms of the elliptic integral of the first kind, <math>~K(\mu)</math>, the modulus of <math>~K</math> can be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2[1+\Chi]^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl[1+\frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr]^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2(\xi_1^2 - 1)^{1/2}}{(\xi_1^2 - 1)^{1/2}+\xi_1} \, .</math>
  </td>
</tr>
</table>
</div>

This is the key result motivating the use of a toroidal coordinate system to evaluate the gravitational potential:  When expressed in an appropriately defined toroidal coordinate system, the modulus of the special function is a function of one, rather than two, spatial coordinates.  This gives some hope that the integral over the second (angular) coordinate, <math>~\xi_2</math>, can be completed analytically, giving rise to an expression for the gravitational potential whose evaluation only requires numerical integration over a single (radial) coordinate, <math>~\xi_1</math>.

Finally, drawing on discussion in [[Appendix/Ramblings/ToroidalCoordinates#ToroidalScaleFactors|our accompanying set of notes]], we recognize that, expressed in terms of toroidal coordinates, the differential area element in the meridional plane is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a^2 \biggl[ \frac{d\xi_1}{(\xi_1 - \xi_2)(\xi_1^2 - 1)^{1/2}} \biggr] 
\biggl[ \frac{d\xi_2}{(\xi_1 - \xi_2)(1-\xi_2^2)^{1/2}} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Putting everything together, then, the (indefinite) integral expression for <math>~q_0</math>, expressed in terms of toroidal-coordinates, is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~q_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int \int \biggl[ \frac{a(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} 
\biggr]^{1/2} \mu K(\mu) \rho(\xi_1, \xi_2) \biggl[ \frac{a}{(\xi_1 - \xi_2)(\xi_1^2 - 1)^{1/2}} \biggr] 
\biggl[ \frac{a}{(\xi_1 - \xi_2)(1-\xi_2^2)^{1/2}} \biggr] d\xi_1 d\xi_2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^{5/2}
\int (\xi_1^2 - 1)^{-1/4} \mu K(\mu) d\xi_1 \int  \rho(\xi_1, \xi_2) 
\biggl[ \frac{d\xi_2}{(\xi_1 - \xi_2)^{5/2}(1-\xi_2^2)^{1/2}} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{1/2} a^{5/2}
\int [ (\xi_1^2 - 1)^{1/2}+\xi_1 ]^{-1/2} K(\mu) d\xi_1 \int  \rho(\xi_1, \xi_2) 
\biggl[ \frac{d\xi_2}{(\xi_1 - \xi_2)^{5/2}(1-\xi_2^2)^{1/2}} \biggr]  \, .
</math>
  </td>
</tr>
</table>
</div>

<!--  OLD WAY
Making the substitution, 
<div align="center">
<math>~\xi_2~ \rightarrow ~ \sin\theta</math> &nbsp; &nbsp; &nbsp; &nbsp;
<math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; &nbsp;
<math>~d\xi_2~ \rightarrow ~ \cos\theta ~d\theta</math> ,
</div>
and, 
<div align="center">
<math>~\xi_1~ \rightarrow ~ \cosh x</math> &nbsp; &nbsp; &nbsp; &nbsp;
<math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; &nbsp;
<math>~d\xi_1~ \rightarrow ~ \sinh x ~dx</math> ,
</div>
gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~q_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{1/2} a^{5/2}
\int\limits_{x_\mathrm{min}}^{x_\mathrm{max}} \frac{K(\mu) \sinh x ~dx}{( \sinh x+\cosh x )^{1/2}}  
\int\limits_{\theta_\mathrm{min}}^{\theta_\mathrm{max}}  \rho(\xi_1, \theta) 
\biggl[ \frac{d\theta}{(\xi_1 - \sin\theta)^{5/2}} \biggr]  \, ,
</math>
  </td>
</tr>
</table>
</div>
where, written in terms of <math>~x</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{2\sinh x}{\sinh x+\cosh x}\biggr]^{1/2}
</math>
  </td>
</tr>
</table>
</div>

and where we have now explicitly introduced four parameters to set definite limits on the nested pair of integrations.
-->

Making the substitution, 
<div align="center">
<math>~\xi_2~ \rightarrow ~ \cos\theta</math> &nbsp; &nbsp; &nbsp; &nbsp;
<math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; &nbsp;
<math>~d\xi_2~ \rightarrow ~ - \sin\theta ~d\theta</math> ,
</div>
gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~q_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-2^{1/2} a^{5/2}
\int\limits_{\xi_1|_\mathrm{min}}^{\xi_1|_\mathrm{max}}  \frac{K(\mu) d\xi_1}{[ (\xi_1^2 - 1)^{1/2}+\xi_1 ]^{1/2} } 
\int\limits_{\theta_\mathrm{min}}^{\theta_\mathrm{max}}  \rho(\xi_1, \theta) 
\biggl[ \frac{d\theta}{(\xi_1 - \cos\theta)^{5/2}} \biggr]  \, ,
</math>
  </td>
</tr>
</table>
</div>

where we have now explicitly introduced four parameters to set definite limits on the nested pair of integrations.  Making the additional substitution, 
<div align="center">
<math>~\xi_1~ \rightarrow ~ \cosh x</math> &nbsp; &nbsp; &nbsp; &nbsp;
<math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; &nbsp;
<math>~d\xi_1~ \rightarrow ~ \sinh x ~dx</math> ,
</div>
gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~q_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 2^{1/2} a^{5/2}
\int\limits_{x_\mathrm{min}}^{x_\mathrm{max}} \frac{K(\mu) \sinh x ~dx}{( \sinh x+\cosh x )^{1/2}}  
\int\limits_{\theta_\mathrm{min}}^{\theta_\mathrm{max}}  \rho(\xi_1, \theta) 
\biggl[ \frac{d\theta}{(\xi_1 - \cos\theta)^{5/2}} \biggr]  \, ,
</math>
  </td>
</tr>
</table>
</div>
where, written in terms of <math>~x</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{2\sinh x}{\sinh x+\cosh x}\biggr]^{1/2} \, .
</math>
  </td>
</tr>
</table>
</div>