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• 14:37, 30 June 2024 Joel2 talk contribs uploaded File:ConstantXi2.png
• 14:36, 30 June 2024 Joel2 talk contribs created page File:Apollonian myway5B.png
• 14:36, 30 June 2024 Joel2 talk contribs uploaded File:Apollonian myway5B.png
• 14:34, 30 June 2024 Joel2 talk contribs created page File:MergedTorus01.png
• 14:34, 30 June 2024 Joel2 talk contribs uploaded File:MergedTorus01.png
• 14:34, 30 June 2024 Joel2 talk contribs created page File:SimpleTorusIllustration.png
• 14:34, 30 June 2024 Joel2 talk contribs uploaded File:SimpleTorusIllustration.png
• 14:32, 30 June 2024 Joel2 talk contribs created page 2DStructure/ToroidalCoordinateIntegrationLimits (Created page with "__FORCETOC__ <!-- will force the creation of a Table of Contents --> <!-- __NOTOC__ will force TOC off --> =Toroidal-Coordinate Integration Limits= In support of our accompanying discussion of the gravitational potential of a uniform-density circular torus, here we explain in detail what limits of integration must be specified in order to accurately determine the ''v...")
• 19:51, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Toroidal08 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions </td> <td align="right"> <math> Q_{-\frac{1}{2}}^\mu(z) + 2\sum_{n=1}^\infty Q^\mu_{n-\frac{1}{2}}(z) \cos(n\nu) </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> e^{i\mu\pi}~\biggl(\frac{\pi}{2} \biggr)^{1 / 2} \Gamma(\mu + \tfrac{1}{2})\b...")
• 19:50, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Toroidal06 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions </td> <td align="right"> <math>(\xi - z)\sum_{m=0}^n (2m+1)P_m(z) Q_m(\xi)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 1 - (\ell+1)[P_{\ell+1}(z) Q_\ell(\xi) - P_\ell(z)Q_{\ell+1}(\xi)] </math> </td> </tr> <tr> <td align="center"...")
• 19:49, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Toroidal05 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions </td> <td align="right"> <math> \int_a^b\biggl[(\nu - \sigma)(\nu + \sigma + 1) + (\rho^2 - \mu^2)(1 - z^2)^{-1} \biggr] w_\nu^\mu ~w_\sigma^\rho ~dz </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \biggl[ z(\nu-\sigma) w_\nu^\mu ~w_\sigm...")
• 19:47, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Toroidal03 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions </td> <td align="right"> <math>Q_\nu^\mu(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> e^{i \mu \pi} ~ (2\pi)^{-\frac{1}{2}} (z^2-1)^{\mu/2} ~\Gamma(\mu + \tfrac{1}{2})~\biggl\{ \int_0^\pi (z - \cos t)^{-\mu - \frac{1}{2}} \cos[(\nu +...")
• 19:13, 29 June 2024 Joel2 talk contribs created page File:RoverDunity.png
• 19:13, 29 June 2024 Joel2 talk contribs uploaded File:RoverDunity.png
• 19:12, 29 June 2024 Joel2 talk contribs created page File:RoverD5over1.png
• 19:12, 29 June 2024 Joel2 talk contribs uploaded File:RoverD5over1.png
• 19:11, 29 June 2024 Joel2 talk contribs created page File:RoverDinfty.png
• 19:11, 29 June 2024 Joel2 talk contribs uploaded File:RoverDinfty.png
• 19:07, 29 June 2024 Joel2 talk contribs created page File:Q0minus1Half.png
• 19:07, 29 June 2024 Joel2 talk contribs uploaded File:Q0minus1Half.png
• 19:00, 29 June 2024 Joel2 talk contribs created page Appendix/Mathematics/ToroidalConfusion (Created page with "<!-- __NOTOC__ will force TOC off --> __FORCETOC__ =Confusion Regarding Whipple Formulae= May, 2018 (J.E.Tohline): I am trying to figure out what the correct relationship is between half-integer degree, associated Legendre functions of the first and second kinds. In order to illustrate my current confusion, here I will restrict my presentation to expressions that give <math>~Q^m_{n - 1 / 2}(\cosh\eta)</math> in terms of <math>~P^n_{m - 1 / 2}(\coth\eta)</math>. ==Pu...")
• 18:58, 29 June 2024 Joel2 talk contribs created page File:ToroidalRecurrenceRelation.png
• 18:58, 29 June 2024 Joel2 talk contribs uploaded File:ToroidalRecurrenceRelation.png
• 18:57, 29 June 2024 Joel2 talk contribs created page File:AbramowitzStegun ToroidalFunctions2.png
• 18:57, 29 June 2024 Joel2 talk contribs uploaded File:AbramowitzStegun ToroidalFunctions2.png
• 18:55, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Toroidal02 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions </td> <td align="right"> <math>Q_{n-1 / 2}^m (\lambda)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>(-1)^n \frac{\pi^{3/2}}{\sqrt{2}~ \Gamma(n-m+1 / 2)} (x^2-1)^{1 / 4} P_{m-1 / 2}^n(x) \, , </math> </td> </tr> <tr> <td align="center...")
• 18:52, 29 June 2024 Joel2 talk contribs created page File:Q0plus3Half.png
• 18:52, 29 June 2024 Joel2 talk contribs uploaded File:Q0plus3Half.png
• 18:52, 29 June 2024 Joel2 talk contribs created page File:Q0plus1Half.png
• 18:52, 29 June 2024 Joel2 talk contribs uploaded File:Q0plus1Half.png
• 18:50, 29 June 2024 Joel2 talk contribs created page File:P0plusHalf.png
• 18:50, 29 June 2024 Joel2 talk contribs uploaded File:P0plusHalf.png
• 18:50, 29 June 2024 Joel2 talk contribs created page File:P0minusHalf.png
• 18:50, 29 June 2024 Joel2 talk contribs uploaded File:P0minusHalf.png
• 18:39, 29 June 2024 Joel2 talk contribs created page Appendix/Mathematics/ToroidalFunctions (Created page with "<!-- __NOTOC__ will force TOC off --> __FORCETOC__ =Relationships Between Toroidal Functions= ==Wong Toroidal Coordinates== ===Wong's Expression for the Potential=== This chapter has been put together in an effort to lay the groundwork for an evaluation of Wong's (1973) derived expression for the gravitational potential both inside and outside of a uniform-density, axisymmetric torus. After multiplying his expression by...")
• 18:34, 29 June 2024 Joel2 talk contribs created page File:Q2minus1Half3.png
• 18:34, 29 June 2024 Joel2 talk contribs uploaded File:Q2minus1Half3.png
• 18:34, 29 June 2024 Joel2 talk contribs created page File:ABSQ1minus1Half3.png
• 18:34, 29 June 2024 Joel2 talk contribs uploaded File:ABSQ1minus1Half3.png
• 18:32, 29 June 2024 Joel2 talk contribs created page File:Q0plus1Half3.png
• 18:32, 29 June 2024 Joel2 talk contribs uploaded File:Q0plus1Half3.png
• 18:31, 29 June 2024 Joel2 talk contribs created page File:P0plus1Half4.png
• 18:31, 29 June 2024 Joel2 talk contribs uploaded File:P0plus1Half4.png
• 18:30, 29 June 2024 Joel2 talk contribs created page File:Q0minus1Half3.png
• 18:30, 29 June 2024 Joel2 talk contribs uploaded File:Q0minus1Half3.png
• 18:28, 29 June 2024 Joel2 talk contribs created page File:P0minus1Half3.png
• 18:28, 29 June 2024 Joel2 talk contribs uploaded File:P0minus1Half3.png
• 18:23, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Toroidal07 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions </td> <td align="right"> <math>P_\nu^{\mu + 1}(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> (z^2-1)^{-\frac{1}{2}} \{ (\nu - \mu) z P^\mu_\nu(z) - (\nu + \mu)P^\mu_{\nu - 1}(z)\} </math> </td> </tr> <tr> <td align="center" colspa...")
• 18:22, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Toroidal04 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/Equation_templates#Relationships_Between_Various_Associated_Legendre_Functions </td> <td align="right"> <math>(\nu - \mu + 1)P^\mu_{\nu + 1} (z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> (2\nu + 1)z P_\nu^\mu(z) - (\nu + \mu)P^\mu_{\nu-1}(z) </math> </td> </tr> <tr> <td align="center" colspan="4"> [https://...")
• 18:17, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Q2minusHalf01 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Toroidal_Function_Evaluations </td> <td align="right"> <math>Q^2_{-\frac{1}{2}}(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{ 4 z E(k) - (z-1) K(k) }{ [2^{3} (z+1) (z-1)^{2} ]^{1 / 2}} </math> </td> </tr> <tr> <td align="center" colspan="4"> where:   <math>k \equiv \sqrt{ \fra...")