Editing 2DStructure/ToroidalCoordinateIntegrationLimits (section)

===Zone I:===
<div id="ZoneI">
<table align="center" border="1" cellpadding="8">
<tr>
  <th align="center" colspan="2"><font size="+1">Figure 4:  </font>&nbsp;Zone I</th>
</tr>
<tr>
  <td align="center">Definition</td>
  <td align="center">Schematic Example</td>
</tr>
<tr>
  <td align="center">
<math>~Z_0 > r_t</math><p></p>for any <math>~a</math>
  </td>
  <td align="center">
[[File:Apollonian_myway5B.png|300px|Apollonian Circles]]
  </td>
</tr>
</table>
</div>

In an [[2DStructure/ToroidalCoordinates#Identifying_Limits_of_Integration|accompanying discussion]], we have derived the following integration limits; numerical values are given for the specific case, <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{3}{4}, \tfrac{3}{4}, \tfrac{1}{4})</math>:

<div align="center" id="LambdaLimits">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Lambda_1 = \xi_1|_\mathrm{max} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2  \biggr]^{-1/2}
\approx 1.1927843</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_1 = \xi_1|_\mathrm{min} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2  \biggr]^{-1/2}
\approx 1.0449467</math>
  </td>
</tr>
</table>

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

where,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\kappa</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
Z_0^2 + a^2  - (\varpi_t^2 - r_t^2) 
</math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~
\frac{5^2}{2^4\cdot 3^2} \approx 0.17361111
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1 + \biggl( \frac{2Z_0}{\kappa}\biggr)^2 ( \varpi_t^2 - r_t^2)
</math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~\frac{17 \cdot 1409}{5^4} \approx 38.3248 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta_\pm</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
- \frac{\kappa}{2} \biggl[ \frac{\varpi_t \mp r_t \sqrt{C}}{(\varpi_t + r_t)(\varpi_t - r_t)} \biggr] 
</math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~
-~\frac{5^2}{2^6\cdot 3}\biggl[ 1\mp \sqrt{ \frac{17\cdot 1409}{3^2\cdot 5^4}} \biggr] 
</math>
  </td>
</tr>
</table>

Notice that we have specified these integration limits such that, in going from the lower limit <math>~(\lambda_1)</math> to the upper limit <math>~(\Lambda_1)</math>, the value of <math>~\xi_1</math> monotonically increases.  <font color="red"><b>CAUTION:</b></font> This statement is often not true.  The quantity, <math>~\kappa</math>, changes signs, depending on whether <math>~(a^2 + Z_0^2) \gtrless (\varpi_t^2 -r_t^2)</math>. When <math>~\kappa</math> changes signs, the two quantities, <math>~\xi_1|_\mathrm{max}</math> and <math>~\xi_1|_\mathrm{min}</math> switch roles; specifically, <math>~\xi_1|_\mathrm{max}</math> becomes less than <math>~\xi_1|_\mathrm{min}</math> (or ''visa versa'').


Also,
<div align="center" id="Parameters">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Gamma_1 = \xi_2\biggr|_+</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_+}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_1 = \xi_2\biggr|_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_-}
</math>
  </td>
</tr>
</table>

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


where, in addition to the quantities defined above,

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

<tr>
  <td align="right">
<math>~\biggl(\frac{\varpi_i}{a}\biggr)_\pm</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\kappa}{2a^2}\cdot \frac{\mathrm{B}}{\mathrm{A}} 
\biggl[1 \pm \sqrt{1-\frac{AC}{B^2}}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{Z_0}{a}\biggr)^2 + \biggl[\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}}\biggr]^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{2\varpi_t Z_0^2}{a\kappa}\biggr) - \biggl[\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}}\biggr] 
</math>
  </td>
</tr>
</table>

Here, our desire also is to specify the integration limits such that <math>~\xi_2</math> monotonically increases in going from the lower limit <math>~(\gamma_1)</math> to the upper limit <math>~(\Gamma_1)</math>.  In order to check to see if this is the case, let's test the limiting values of  <math>~\xi_2</math> when we are considering a radial-coordinate value roughly midway between its limits, say, when <math>~\xi_1 = 1.1</math>.  For this specific case, we find,
<div align="center">
<math>~\frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} = 2.400397 \, ;</math><p></p>
<math>~\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}} = - 0.150397 \, ;</math><p></p>
<math>~A = 5.085119 \, ;</math><p></p>
<math>~B = 14.73040 \, ;</math><p></p>
<math>~\biggl(\frac{\varpi_i}{a}\biggr)_\pm = 2.263098\biggl[1 \pm \sqrt{1-0.898157}  \biggr] \, ;</math><p></p>
<math>~\Rightarrow ~~~ \biggl(\frac{\varpi_i}{a}\biggr)_+ = 2.985318 \, ;</math><p></p>
<math>~\Rightarrow ~~~ \biggl(\frac{\varpi_i}{a}\biggr)_- = 1.540878 \, ;</math><p></p>
<math>~\Rightarrow ~~~ \xi_2\biggr|_+ = 0.946496 \, ;</math><p></p>
<math>~\Rightarrow ~~~ \xi_2\biggr|_- = 0.802600 \, .</math>
</div>
Hence, our ordering of the limits appears to be the one desired.