User contributions for Joel2

30 June 2024

• 14:4514:45, 30 June 2024 diff hist 0 N File:TorusTest003Sm.png No edit summary current
• 14:4414:44, 30 June 2024 diff hist 0 N File:TorusTest001Sm.png No edit summary current
• 14:4314:43, 30 June 2024 diff hist 0 N File:TorusTest000Sm.png No edit summary current
• 14:4214:42, 30 June 2024 diff hist 0 N File:LimitsOnTorus2ColoredSmall01.png No edit summary current
• 14:4114:41, 30 June 2024 diff hist 0 N File:Apollonian myway8B.png No edit summary current
• 14:4014:40, 30 June 2024 diff hist 0 N File:Xi2Movie04.gif No edit summary current
• 14:3914:39, 30 June 2024 diff hist 0 N File:Torus z15B.png No edit summary current
• 14:3814:38, 30 June 2024 diff hist 0 N File:Zone II08.png No edit summary current
• 14:3814:38, 30 June 2024 diff hist 0 N File:Apollonian myway7B.png No edit summary current
• 14:3714:37, 30 June 2024 diff hist 0 N File:ConstantXi2.png No edit summary current
• 14:3614:36, 30 June 2024 diff hist 0 N File:Apollonian myway5B.png No edit summary current
• 14:3414:34, 30 June 2024 diff hist 0 N File:MergedTorus01.png No edit summary current
• 14:3414:34, 30 June 2024 diff hist 0 N File:SimpleTorusIllustration.png No edit summary current
• 14:3214:32, 30 June 2024 diff hist +112,068 N 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..." current

29 June 2024

• 19:5119:51, 29 June 2024 diff hist +956 N 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..." current
• 19:5019:50, 29 June 2024 diff hist +760 N 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"..." current
• 19:4919:49, 29 June 2024 diff hist +1,019 N 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..." current
• 19:4719:47, 29 June 2024 diff hist +1,154 N 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 +..." current
• 19:1319:13, 29 June 2024 diff hist 0 N File:RoverDunity.png No edit summary current
• 19:1219:12, 29 June 2024 diff hist 0 N File:RoverD5over1.png No edit summary current
• 19:1119:11, 29 June 2024 diff hist 0 N File:RoverDinfty.png No edit summary current
• 19:0719:07, 29 June 2024 diff hist 0 N File:Q0minus1Half.png No edit summary current
• 19:0019:00, 29 June 2024 diff hist +25,227 N 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..." current
• 18:5818:58, 29 June 2024 diff hist 0 N File:ToroidalRecurrenceRelation.png No edit summary current
• 18:5718:57, 29 June 2024 diff hist 0 N File:AbramowitzStegun ToroidalFunctions2.png No edit summary current
• 18:5518:55, 29 June 2024 diff hist +799 N 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..." current
• 18:5218:52, 29 June 2024 diff hist 0 N File:Q0plus3Half.png No edit summary current
• 18:5218:52, 29 June 2024 diff hist 0 N File:Q0plus1Half.png No edit summary current
• 18:5018:50, 29 June 2024 diff hist 0 N File:P0plusHalf.png No edit summary current
• 18:5018:50, 29 June 2024 diff hist 0 N File:P0minusHalf.png No edit summary current
• 18:3918:39, 29 June 2024 diff hist +82,553 N 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..." current
• 18:3418:34, 29 June 2024 diff hist 0 N File:Q2minus1Half3.png No edit summary current
• 18:3418:34, 29 June 2024 diff hist 0 N File:ABSQ1minus1Half3.png No edit summary current
• 18:3218:32, 29 June 2024 diff hist 0 N File:Q0plus1Half3.png No edit summary current
• 18:3118:31, 29 June 2024 diff hist 0 N File:P0plus1Half4.png No edit summary current
• 18:3018:30, 29 June 2024 diff hist 0 N File:Q0minus1Half3.png No edit summary current
• 18:2818:28, 29 June 2024 diff hist 0 N File:P0minus1Half3.png No edit summary current
• 18:2318:23, 29 June 2024 diff hist +960 N 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..." current
• 18:2218:22, 29 June 2024 diff hist +944 N 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://..." current
• 18:1718:17, 29 June 2024 diff hist +669 N 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..." current
• 18:1518:15, 29 June 2024 diff hist +52,353 N Appendix/Mathematics/ToroidalSynopsis01 Created page with "<!-- __FORCETOC__ will force the creation of a Table of Contents --> <!-- __NOTOC__ will force TOC off --> =Synopsis of Toroidal Coordinate Approach= ==Basics== Here we attempt to bring together — in as succinct a manner as possible — our approach and [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong's (1973)] approach to determining the gravi..." current
• 18:1418:14, 29 June 2024 diff hist +646 N Template:Math/EQ Q1minusHalf01 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^1_{-\frac{1}{2}}(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \biggl[\frac{1}{2(z-1)} \biggr]^{1 / 2} E(k) </math> </td> </tr> <tr> <td align="center" colspan="4"> where:   <math>k = \sqrt{ \frac{2}{z+1}} \, .</math>..." current
• 18:1218:12, 29 June 2024 diff hist +763 N Template:Math/EQ QplusHalf01 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_{+\frac{1}{2}}(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> z \sqrt{ \frac{2}{z+1} }~K\biggl( \sqrt{ \frac{2}{z+1} } \biggr) ~-~ [2(z+1)]^{1 / 2} E\biggl( \sqrt{ \frac{2}{z+1} } \biggr) </math> </td> </tr> <tr> <td align="cente..." current
• 18:0718:07, 29 June 2024 diff hist +1,361 N Template:Math/EQ PplusHalf01 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>P_{+ \frac{1}{2}}(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2}{\pi} \biggl[z + \sqrt{ z^2-1} \biggr]^{1 / 2} ~E\biggl( \sqrt{ \frac{2(z^2-1)^{1 / 2}}{z + (z^2-1)^{1 / 2}}} \biggr) </math> </td> <td align="center" rowspa..." current
• 18:0218:02, 29 June 2024 diff hist +1,272 N Template:Math/EQ QminusHalf01 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_{-\frac{1}{2}}(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \sqrt{ \frac{2}{z+1} } ~K\biggl( \sqrt{ \frac{2}{z+1}} \biggr) </math> </td> <td align="center" rowspan="2" width="20%">      for example …</td>..." current
• 18:0118:01, 29 June 2024 diff hist +1,374 N Template:Math/EQ PminusHalf01 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>P_{-\frac{1}{2}}(z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \frac{2}{\pi} \biggl[\frac{2}{z+1}\biggr]^{1 / 2} ~K\biggl( \sqrt{ \frac{z-1}{z+1}} \biggr) </math> </td> <td align="center" rowspan="2" width="20%">    &nb..." current
• 17:5917:59, 29 June 2024 diff hist +584 N Template:Math/EQ Toroidal00 Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=http:Appendix/Equation_templates#Special_Function_Relationships </td> <td align="right"> <math>~ P_\nu^n(z) </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{\Gamma(\nu + n + 1)}{\Gamma(\nu - n + 1)} P_\nu^{-n}(z) </math> </td> </tr> <tr> <td align="center" colspan=4> [https://authors.library.caltech.edu/43491/1/V..." current
• 17:5717:57, 29 June 2024 diff hist +914 N Template:Math/EQ EllipticIntegral04 Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Complete_Elliptic_Integrals </td> <td align="right"> <math>E(\mu)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 1 ~+~ \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{1\cdot 2}\biggr)(k')^2 ~+~ \frac{1^2\cdot 3}{2^2\cdot 4}\biggl( \ln \frac{4}{k'} - \frac{2}{1\cdot 2} - \frac{1}{3\cdot 4}\biggr)(k')..." current
• 17:5517:55, 29 June 2024 diff hist +757 N Template:Math/EQ EllipticIntegral02 Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Complete_Elliptic_Integrals </td> <td align="right"> <math>\frac{2E(k)}{\pi}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 1 - \frac{1}{2^2} ~k^2 - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 - \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5} - \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^..." current
• 17:5417:54, 29 June 2024 diff hist +954 N Template:Math/EQ EllipticIntegral03 Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Complete_Elliptic_Integrals </td> <td align="right"> <math>K(\mu)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> \ln \frac{4}{k^'} + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} \biggr)(k')^2 + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 \biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} -..." current