View source for User:Tohline/Appendix/Mathematics/ToroidalFunctions

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=Relationships Between Toroidal Functions=

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=Drawn From Discussion of Solving the Poisson Equation=
The following has been copied (May 2018) from an [[User:Tohline/AxisymmetricConfigurations/PoissonEq#Toroidal_Functions|accompanying chapter]] that presents the ''integral representation'' of the Poisson equation in terms of toroidal functions.

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<th align="center"><font size="+0">Table 5: &nbsp;Green's Function in Terms of<br />Zero Order, Half-(Odd)Integer Degree, Associated Legendre Functions of the Second Kind, <math>~Q^0_{m-1 / 2}(\chi)</math><br />(also referred to as Toroidal Functions)</font></th>
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<math>~ \frac{1}{|\vec{x} - \vec{x}^{~'}|}</math>
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<math>~=</math>
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<math>~ 
\frac{1}{\pi \sqrt{\varpi \varpi^'}} \sum_{m=-\infty}^{\infty} e^{im(\phi - \phi^')}Q_{m- 1 / 2}(\chi)
</math>
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where:<br />
<div align="center"><math>~\chi \equiv \frac{\varpi^2 + (\varpi^')^2 + (z - z^')^2}{2\varpi \varpi^'}</math><br /><br />
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl &amp; J. E. Tohline (1999)], p. 88, Eqs. (15) &amp; (16)<br />
See also the [https://dlmf.nist.gov/14.19#ii DLMF's definition of Toroidal Functions], <math>~Q_{m - 1 / 2}^{0}</math>
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Note that, according to, for example, equation (8.731.5) of Gradshteyn &amp; Ryzhik (1994), 
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<math>~Q^0_{-m - 1 / 2}(\chi) = Q^0_{m- 1 / 2}(\chi) \, .</math>
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Hence, the Green's function can straightforwardly be rewritten in terms of a simpler summation over just ''non-negative'' values of the index, <math>~m</math>.
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Referencing equations (8.13.3) and (8.13.7), respectively, of Abramowitz &amp; Stegun (1965), we see that for the smallest two values of the ''non-negative'' index, <math>~m</math>, the function, <math>~Q_{m- 1 / 2}(\chi)</math>, can be rewritten in terms of, the more familiar, complete elliptic integrals of the first and second kind.  Specifically, 
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for <math>~m = 0</math>,
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<math>~Q_{m- 1 / 2}(\chi) ~\rightarrow~ Q_{-1 / 2}</math>
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<math>~=</math>
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<math>~
\mu K(\mu) \, ,
</math>
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and, for <math>~m = 1</math>,
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<math>~Q_{m- 1 / 2}(\chi) ~\rightarrow~ Q_{+ 1 / 2}</math>
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<math>~=</math>
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<math>~
\chi \mu K(\mu) - (1+\chi)\mu E(\mu) \, ,
</math>
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where,
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<math>~\mu \equiv \biggl[ \frac{2}{1+\chi} \biggr]^{1 / 2}</math>
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<math>~=</math>
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<math>~
\biggl[ \frac{4\varpi \varpi^'}{(\varpi + \varpi^')^2 + (z - z^')^2} \biggr]^{1 / 2}
\, .
</math>
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Excerpt from p. 337 of [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 M. Abramowitz &amp; I. A. Stegun (1995)]
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[[File:AbramowitzStegun ToroidalFunctions2.png|center|700px|Abramowitz &amp; Stegun (1965)]]
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Finally, equation (8.5.3) from Abramowitz &amp; Stegun (1965) or equation (8.832.4) of Gradshteyn &amp; Ryzhik (1994) &#8212; also see equation (2) of [http://adsabs.harvard.edu/abs/2000JCoPh.161..204G Gil, Segura &amp; Temme (2000)] &#8212; provide the recurrence relation for all other values of the index, <math>~m</math>.  Specifically, for all <math>~m \ge 2</math>,
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<math>~Q_{m - 1 / 2}(\chi) = 4\biggl[\frac{m-1}{2m-1}\biggr] \chi Q_{m- 3 / 2}(\chi) 
- \biggl[ \frac{2m-3}{2m-1}\biggr] Q_{m- 5 / 2}(\chi) \, .</math>
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Excerpt from p. 490 of [https://dl-acm-org.libezp.lib.lsu.edu/citation.cfm?id=365474&picked=prox W. Guatschi (1965, Communications of the ACM, 8, 488 - 492)]
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[[File:ToroidalRecurrenceRelation.png|center|500px|Guatschi (1965, Communications of the ACM, 8, 488 - 492)]]
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=See Also=


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