Editing Apps/Wong1973Potential (section)

=====Analytic Integration Over the Angular Coordinate=====
However, focusing only on the integration over the angular coordinate, we see that the integrand in the expression for <math>\Phi_\mathrm{W}</math> is significantly less imposing than the one that appears in the expression for <math>\Phi_\mathrm{CT}</math>.  {{ Wong73 }} was able to evaluate this definite integral in closed form, analytically.  While Wong does not record the detailed steps that he used to evaluate this definite integral, he does indicate that he received guidance from Volume I of [https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erd&eacute;lyi's (1953)] ''Higher Transcendental Functions''.  We therefore presume that he adopted the line of reasoning that we have [[#A.3|detailed in the Appendix, below, in deriving the expression labeled]] <font color="green" size="+1">&#x2462;</font>.  Wong recognized, what we have explicitly demonstrated, namely,
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<math>\int_{-\pi}^\pi d\theta^' (\cosh\eta^' - \cos\theta^')^{- 5 / 2} \cos[n(\theta - \theta^')]</math>
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<math>=</math>
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<math>2\cos(n\theta)
\int_0^\pi \frac{ \cos(n\theta^')~d\theta^' }{ (\cosh\eta^' - \cos \theta^')^{\frac{5}{2}} }
</math>
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<math>=</math>
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<math>
\frac{8\sqrt{2}}{3} \biggl[ \frac{\cos(n\theta)}{\sinh^2\eta^'} \biggr] Q^2_{n- \frac{1}{2}} (\cosh\eta^') \,.
</math>
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{{ Wong73 }}, p. 293, Eq. (2.56)
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<font color="red">'''CAUTION:'''</font>&nbsp; It is important to appreciate that, in this expression as well as in the expressions to follow, the term, <math>Q^2_{n-\frac{1}{2}}(z)</math>, is ''not'' the square of the zero-order toroidal function, <math>Q^0_{n - \frac{1}{2}}(z)</math>, but is instead the toroidal function of order two.  In an [[Appendix/Mathematics/ToroidalSynopsis01#Evaluating_Q2.CE.BD|accompanying discussion]] we present an analytic expression for <math>Q^2_{-\frac{1}{2}}(z)</math> &#8212; and a separate analytic expression for <math>Q^1_{-\frac{1}{2}}(z)</math> &#8212; in terms of complete elliptic integrals, as well as a recurrence relation that can be used to generate analytic expressions for all other order-two (and all other order-one) toroidal functions that have higher half-integer degrees, <math>n-\tfrac{1}{2}</math> for <math>n \ge 1</math>.
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Hence, Wong was able to simplify the expression for <math>\Phi_\mathrm{W}</math> to one that &#8212; albeit, in addition to an infinite summation over the index, <math>n</math> &#8212; only requires integration over the radial coordinate, <math>\eta^'</math>.  Specifically, he obtained, 

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<math>\Phi_\mathrm{W}(\eta,\theta)</math>
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<math>=</math>
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<math>
- 2G \rho_0 a^2 (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n 
\int_{\eta_0}^\infty d\eta^' ~\sinh\eta^'~P_{n-1 / 2}(\cosh\eta_<) ~Q_{n-1 / 2}(\cosh\eta_>)\biggl\{
\frac{8\sqrt{2}}{3} \biggl[ \frac{\cos(n\theta)}{\sinh^2\eta^'} \biggr] Q^2_{n- \frac{1}{2}} (\cosh\eta^')
\biggr\}
</math>
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&nbsp;
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<math>=</math>
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<math>
- \biggl( \frac{16\sqrt{2}}{3}  \biggr) G \rho_0 a^2 (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta)
\int_{\eta_0}^\infty d\eta^' \Biggl[ \frac{Q^2_{n- \frac{1}{2}} (\cosh\eta^')}{\sinh\eta^'} \Biggr] ~P_{n-1 / 2}(\cosh\eta_<) ~Q_{n-1 / 2}(\cosh\eta_>)
\, .
</math>
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{{ Wong73 }}, p. 294, Eq. (2.57)
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