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17:23, 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...")
17:22, 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://...")
17:17, 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...")
17:15, 29 June 2024 Joel2 talk contribs created page Appendix/Mathematics/ToroidalSynopsis01 (Created page with "  =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...")
17:14, 29 June 2024 Joel2 talk contribs created page 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>...")
17:12, 29 June 2024 Joel2 talk contribs created page 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...")
17:07, 29 June 2024 Joel2 talk contribs created page 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...")
17:02, 29 June 2024 Joel2 talk contribs created page 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>...")
17:01, 29 June 2024 Joel2 talk contribs created page 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...")
16:59, 29 June 2024 Joel2 talk contribs created page 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...")
16:57, 29 June 2024 Joel2 talk contribs created page 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')...")
16:55, 29 June 2024 Joel2 talk contribs created page 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^...")
16:54, 29 June 2024 Joel2 talk contribs created page 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} -...")
16:53, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ EllipticIntegral01 (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{2K(k)}{\pi}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 1 + \biggl( \frac{1}{2} \biggr)^2k^2 + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 + \biggl( \frac{1\cdot 3\cdot...")
16:50, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Gamma01 (Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Gamma_Function </td> <td align="right" colspan="2"> <math>~ \Gamma(z) ~\Gamma(1-z) </math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \frac{\pi}{\sin(\pi z)} </math> </td> <td align="center" rowspan="1"><math>~\biggl|</math></td> <td align="center" rowspan="3"> for example, if<br /> <math>~z \...")
16:48, 29 June 2024 Joel2 talk contribs created page Appendix/SpecialFunctions (Created page with "__FORCETOC__  =50pxSpecial Functions= ===Gamma Function=== <div align="center"> <table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue"> <tr> <th colspan=3 align="center"> <font size="+1" color="darkblue">Gamma Function</font> </th> </tr> <tr> <td colspan=2> To insert a given equation into any Wiki document, type ...<br /><center> {{ Math/<i><font color="red">Template_Name</fo...")
16:43, 29 June 2024 Joel2 talk contribs created page Template:Math/EQ Toroidal01 (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[t t^' - (t^2-1)^{1 / 2} (t^{'2} - 1)^{1 / 2} \cos\psi] </math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> Q_\nu(t) P_\nu(t^') + 2\sum_{n=1}^\infty (-1)^n Q^n_\nu(t) P^{-n}_\nu(t^') \cos(n\psi) </math>...")
14:12, 29 June 2024 Joel2 talk contribs created page File:DysonTorusIllustration03.png
14:12, 29 June 2024 Joel2 talk contribs uploaded File:DysonTorusIllustration03.png
14:00, 29 June 2024 Joel2 talk contribs created page Apps/DysonPotential (Created page with "__FORCETOC__   =Dyson (1893)= {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:#ffeeee;" |<b>Dyson<br />(1893)</b> |} Our focus, here, is on the pioneering work of [http://adsabs.harvard.edu/abs/1893RSPTA.184....")
13:57, 29 June 2024 Joel2 talk contribs created page File:FukushimaTorusIllustration02.png
13:57, 29 June 2024 Joel2 talk contribs uploaded File:FukushimaTorusIllustration02.png
13:56, 29 June 2024 Joel2 talk contribs created page File:WongTorusIllustration02.png
13:56, 29 June 2024 Joel2 talk contribs uploaded File:WongTorusIllustration02.png
13:55, 29 June 2024 Joel2 talk contribs created page File:OstrikerTorusIllustration01.png
13:55, 29 June 2024 Joel2 talk contribs uploaded File:OstrikerTorusIllustration01.png
13:55, 29 June 2024 Joel2 talk contribs created page File:MacMillanTorusIllustration02.png
13:55, 29 June 2024 Joel2 talk contribs uploaded File:MacMillanTorusIllustration02.png
13:53, 29 June 2024 Joel2 talk contribs created page File:DysonTorusIllustration04.png
13:53, 29 June 2024 Joel2 talk contribs uploaded File:DysonTorusIllustration04.png
13:49, 29 June 2024 Joel2 talk contribs created page Appendix/Ramblings/CCGF (Created page with "__FORCETOC__   =Compact Cylindrical Green Function (CCGF)= ==Preface by Tohline== [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)] present an expression for the Newtonian gravitational potential in terms of a ''Compact Cylindrical Green's Function'' expansion. Over a professional career that dates back to 1976, this has turned out to be one of my mos...")
13:47, 29 June 2024 Joel2 talk contribs created page Apps/DWT (Created page with "__FORCETOC__  =Common Theme: Determining the Gravitational Potential for Axisymmetric Mass Distributions= You have arrived at this page from our ''Tiled Menu'' by clicking on the chapter title that is also identified, below, by the light-blue-colored icon. You may proceed directly to that chapter by clicking (again) on that highlighted chapter icon. However, we have brought y...")
18:50, 28 June 2024 Xxsrm talk contribs created page Appendix/Ramblings/PowerSeriesExpressions (Created page with "__FORCETOC__  =Approximate Power-Series Expressions= ==Broadly Used Mathematical Expressions (shown here without proof)== ===Binomial=== <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~(1 \pm x)^n</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~\pm ~nx + \biggl[\frac{n(n-1)}{2!}\biggr]x^2 ~\pm~ \biggl[\frac{n(n-1)(n-2)}{3!}\biggr]x^3...")
18:29, 28 June 2024 Xxsrm talk contribs created page File:WolframAlphaResult.png
18:29, 28 June 2024 Xxsrm talk contribs uploaded File:WolframAlphaResult.png
18:26, 28 June 2024 Xxsrm talk contribs created page File:Ostriker64PaperIIFig1.png
18:26, 28 June 2024 Xxsrm talk contribs uploaded File:Ostriker64PaperIIFig1.png
18:21, 28 June 2024 Xxsrm talk contribs created page Apps/Ostriker64 (Created page with "__FORCETOC__  =Polytropic & Isothermal Tori= {| class="Ostriker64" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:white;" |<b>Ostriker<br />(1964)</b> |} Here we will focus on the analysis of the structure self-gravitating tori that are composed of compressible — speci...")
18:03, 28 June 2024 Xxsrm talk contribs created page File:EH85ten.png
18:03, 28 June 2024 Xxsrm talk contribs uploaded File:EH85ten.png
18:03, 28 June 2024 Xxsrm talk contribs created page File:EH85nine.png
18:03, 28 June 2024 Xxsrm talk contribs uploaded File:EH85nine.png
18:01, 28 June 2024 Xxsrm talk contribs created page File:MPT77eight.png
18:01, 28 June 2024 Xxsrm talk contribs uploaded File:MPT77eight.png
18:01, 28 June 2024 Xxsrm talk contribs created page File:MPT77seven.png
18:01, 28 June 2024 Xxsrm talk contribs uploaded File:MPT77seven.png
17:58, 28 June 2024 Xxsrm talk contribs created page File:MPT77sixModified.png
17:58, 28 June 2024 Xxsrm talk contribs uploaded File:MPT77sixModified.png
17:58, 28 June 2024 Xxsrm talk contribs created page File:MPT77fiveModified.png
17:58, 28 June 2024 Xxsrm talk contribs uploaded File:MPT77fiveModified.png

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