Editing 2DStructure/ToroidalGreenFunction (section)

==Green's Function Expression==

===As presented by Wong (1973)===
Referencing [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>], [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] states that, in toroidal coordinates, the Green's function is,
<div align="center">
<table border="0" cellpadding="5" 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{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 }
\sum\limits^\infty_{m,n=0} (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \cos[m(\psi - \psi^')]\cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta
\end{cases}\, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], p. 293, Eq. (2.53)<br />
[see also: [http://adsabs.harvard.edu/abs/1997JMP....38.3679B J. W. Bates (1997)], p. 3685, Eq. (31)]<br />
[see also: [http://adsabs.harvard.edu/abs/2000AN....321..363C Cohl, Tohline, Rau, &amp; Srivastava (2000)], &sect;6.2, Eq. (48)]
  </td>
</tr>
</table>
</div>
where, <math>~P^m_{n-1 / 2}, Q^m_{n-1 / 2}</math> are ''Associated Legendre Functions'' of the first and second kind with degree <math>~n - \tfrac{1}{2}</math> and order <math>~m</math> (toroidal harmonics), and <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0 = 1</math> and <math>~\epsilon_m = 2</math> for all <math>~m \ge 1</math>.  This Green's function expression can indeed be found as eq. (10.3.81) on p. 1304 of [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>], but it should be noted that the MF53 expression differs from Wong's in two respects (see footnote 2 on p. 370 of [http://adsabs.harvard.edu/abs/2000AN....321..363C Cohl et al. (2000)] for a proposed explanation):  First, the factor, <math>~(-1)^m</math>, appears as <math>~(-i)^m</math> in MF53; and, second, in the term that is composed of a ratio of gamma functions, the denominator appears in MF53 as <math>~\Gamma(n - m + \tfrac{1}{2})</math>, whereas it should be <math>~\Gamma(n + m + \tfrac{1}{2})</math>, as presented here.
<!--
For later reference, note that after drawing from, for example, [https://en.wikipedia.org/wiki/Gamma_function#General Wikipedia's account of the general properties of gamma functions],  the collection of factors immediately inside the double summation may be more explicitly written as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~L_n^m \equiv (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(-1)^m \epsilon_m \epsilon_n \biggl[ \frac{\sqrt{\pi}[2(n-m)]!}{4^{n-m}(n-m)!} \biggr]
\biggl[ \frac{4^{n+m}(n+m)!}{\sqrt{\pi}[2(n+m)]!} \biggr]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(-1)^m \epsilon_m \epsilon_n \biggl[ \frac{\cdots [2(n-m)-4]\cdot [2(n-m)-3]\cdot[2(n-m)-2]\cdot[2(n-m)-1]\cdot[2(n-m)] }{\cdots [(n-m)-4]\cdot [(n-m)-3]\cdot [(n-m)-2]\cdot [(n-m)-1]\cdot (n-m) } \biggr]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(-1)^m \epsilon_m \epsilon_n \biggl[ \frac{\cdots 2[(n-m)-2]\cdot [2(n-m)-3]\cdot 2[(n-m)-1]\cdot [2(n-m)-1] \cdot 2(n-m) }{\cdots [(n-m)-4]\cdot [(n-m)-3]\cdot [(n-m)-2]\cdot [(n-m)-1]\cdot (n-m) } \biggr]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(-1)^m \epsilon_m \epsilon_n \biggl[ \frac{\cdots 2^{n-m}  [2(n-m)-3]\cdot  [2(n-m)-1] }{1} \biggr]
\biggl[ \frac{1}{\cdots 2^{n+m}  [2(n+m)-3]\cdot [2(n+m)-1]] } \biggr]2^{4m}
</math>
  </td>
</tr>
</table>
-->
===As Presented by Hicks (1881)===
Notably, more than 135 years ago, [http://rstl.royalsocietypublishing.org/content/172/609.full.pdf+html W. M. Hicks (1881)] already had constructed a Green's function expression for the reciprocal distance between two points in toroidal coordinates.   The following boxed-in table contains a snapshot image of  equation (35) from Hicks (1881), along with its associated text; notice that he refers to <math>~\phi</math> as the "potential."  Although notations are different &#8212; for example, <math>~C</math> is shorthand for <math>~\cosh u</math> and <math>~c</math> is shorthand for <math>~\cos v</math> &#8212; one can easily see factor-by-factor agreement when compared with the Green's function [[#As_presented_by_Wong_.281973.29|presented by Wong (1973)]]. 
<table border="1" cellpadding="10" align="center">
<tr>
  <td align="center">
<!-- [[File:Hicks1881TitlePage.png|500px|Title Page of Hicks (1881)]] -->
[https://ui.adsabs.harvard.edu/abs/1881RSPT..172..609H/abstract W. M. Hicks (1881)]<br />
''"On Toroidal Functions"''<br />
Philosophical Transactions of the Royal Society of London, vol. 172, pp. 609-652
</td>
</tr>
<tr><td align="center">
[[File:Hicks1881GreenFunction.png|750px|To be inserted:  Eq. (35) from Hicks (1881)]]
</td></tr></table>

===Rearranging Terms and Incorporating Special-Function Relations===

Let's focus on the situation when <math>~\eta^' > \eta</math>, and begin rearranging or substituting terms.
<div align="center">
<table border="0" cellpadding="5" 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{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 }
\sum\limits^\infty_{m=0} (-1)^m \epsilon_m \cos[m(\psi - \psi^')]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sum\limits^\infty_{n=0} \epsilon_n \cos[n(\theta - \theta^')]  ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})}
~P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^')
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} }
\sum\limits^\infty_{m=0} (-1)^m \epsilon_m \cos[m(\psi - \psi^')]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sum\limits^\infty_{n=0} \epsilon_n \cos[n(\theta - \theta^')] \biggl\{ ~
\sqrt{ \frac{\pi}{2} }~\Gamma(n-m+\tfrac{1}{2}) \sqrt{\sinh\eta}~P^m_{n-1 / 2}(\cosh\eta) 
\biggl\}\biggr\{
~\sqrt{ \frac{2}{\pi} }~\frac{\sqrt{\sinh\eta^'}}{\Gamma(n + m + \tfrac{1}{2})} Q^m_{n-1 / 2}(\cosh\eta^')
\biggr\}
</math>
  </td>
</tr>
</table>
</div>

The term contained within the first set of curly braces on the right-hand side of this expression can be replaced by the derived expression labeled <font color="green" size="+1">&#x2460;</font> in the [[#A.1|Appendix, below]], and simultaneously the term contained within the second set of curly braces can be replaced by the derived expression labeled <font color="green" size="+1">&#x2461;</font> in the [[#A.2|same Appendix]].  After making these substitutions, we have,
<table border="0" cellpadding="5" 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{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} }
\sum\limits^\infty_{m=0} (-1)^m \epsilon_m \cos[m(\psi - \psi^')]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sum\limits^\infty_{n=0} \epsilon_n \cos[n(\theta - \theta^')] \biggl\{ ~
(-1)^{-n}Q^n_{m-1 / 2}(\coth\eta) 
\biggl\}\biggr\{
~(-1)^{-m} P^{-n}_{m-1 / 2}(\coth\eta^')
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} }
\sum\limits^\infty_{m=0} \epsilon_m \cos[m(\psi - \psi^')]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \sum\limits^\infty_{n=0} (-1)^{n} \epsilon_n \cos[n(\theta - \theta^')] 
Q^n_{m-1 / 2}(\coth\eta) P^{-n}_{m-1 / 2}(\coth\eta^') \, ,
</math>
  </td>
</tr>
</table>

where, in writing this last expression we have acknowledged that, since <math>~n</math> is either zero or a positive integer, <math>~(-1)^{-n} = (-1)^n</math>.  Next we draw upon the "Key Equation" relation,

{{ Math/EQ_Toroidal01 }} 

which, after making the substitutions, <math>~\nu \rightarrow (m - \tfrac{1}{2})</math> and <math>~\psi \rightarrow (\theta - \theta^')</math>, and incorporating the Neumann factor, <math>~\epsilon_n</math>, becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
Q_{m - \frac{1}{2} }\ [t t^' - (t^2-1)^{1 / 2} (t^{'2} - 1)^{1 / 2} \cos(\theta- \theta^') ]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sum_{n=0}^\infty (-1)^n \epsilon_n Q^n_{m - \frac{1}{2} }(t) P^{-n}_{m - \frac{1}{2} }(t^') \cos[(n(\theta- \theta^')] \, .
</math>
  </td>
</tr>
</table>

Finally, after making the associations, <math>~t \rightarrow \coth\eta</math> and <math>~t^' \rightarrow \coth\eta^'</math>, this last expression allows us to rewrite Wong's (1973) Green's function in a significantly more compact form, namely,
<table border="0" cellpadding="5" 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{ [ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^')]^{1 /2 } }{\pi a \sqrt{\sinh\eta^'} \sqrt{\sinh\eta} }
\sum\limits^\infty_{m=0} \epsilon_m \cos[m(\psi - \psi^')]  Q_{m - \frac{1}{2}}(\Chi) \, ,
</math>
  </td>
</tr>
</table>
where the argument, <math>~\Chi</math>, of the toroidal function, <math>~Q_{m - \frac{1}{2}}</math>, is,

<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>~
t t^' - (t^2-1)^{1 / 2} (t^{'2} - 1)^{1 / 2} \cos(\theta- \theta^') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\coth\eta \coth\eta^' - (\coth^2\eta-1)^{1 / 2} (\coth^2\eta'- 1)^{1 / 2} \cos(\theta- \theta^') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\cosh\eta \cosh\eta^'}{\sinh\eta \sinh\eta^'} - \biggl[ \frac{1}{\sinh^2\eta} \biggr]^{1 / 2} \biggl[ \frac{1}{\sinh^2\eta'}\biggr]^{1 / 2} \cos(\theta- \theta^') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\cosh\eta \cosh\eta^' -  \cos(\theta- \theta^') }{\sinh\eta \sinh\eta^'} \, .
</math>
  </td>
</tr>
</table>

===As Presented in Cohl &amp; Tohline (1999)===

This last, compact Green's function expression &#8212; which we have derived, here, from [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] published Green's function by drawing strategically upon a variety of special function relations &#8212; precisely matches the "compact cylindrical Green's function expression" that has been derived independently by [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)] via a less tortuous route, namely,
<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=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) 
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="2">&nbsp;</td>
  <td align="left" colspan="1">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], p. 88, Eq. (17)
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{a\pi} 
\biggl[  \frac{(\cosh\eta^' - \cos\theta^')}{\sinh\eta^' } \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2}
\sum_{m=0}^{\infty} \epsilon_m \cos[m(\psi - \psi^')] Q_{m- 1 / 2}(\Chi) \, ,
</math>
  </td>
</tr>
</table>
where,
<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{(\varpi^')^2 + \varpi^2 + (z^' - z)^2}{2\varpi^' \varpi}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^'} \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="2">
&nbsp; &nbsp;
  </td>
  <td align="left" colspan="3">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], p. 88, Eq. (16)
  </td>
</tr>
</table>


'''Note from J. E. Tohline (June, 2018)''':  &nbsp;This is the first time that I have been able to formally demonstrate to myself that these two separately derived Green's function expressions are identical.  See, however, the earlier identification of ''new'' addition theorems in association with equations (49) and (50) of [http://adsabs.harvard.edu/abs/2000AN....321..363C Cohl et al. (2000)].