Main public logs

Combined display of all available logs of JETohlineWiki. You can narrow down the view by selecting a log type, the username (case-sensitive), or the affected page (also case-sensitive).

• 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...")
• 18:15, 29 June 2024 Joel2 talk contribs created page 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...")
• 18:14, 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>...")
• 18:12, 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...")
• 18:07, 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...")
• 18:02, 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>...")
• 18:01, 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...")
• 17:59, 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...")
• 17:57, 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')...")
• 17:55, 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^...")
• 17:54, 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} -...")
• 17:53, 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...")
• 17:50, 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 \...")
• 17:48, 29 June 2024 Joel2 talk contribs created page Appendix/SpecialFunctions (Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =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...")
• 17:43, 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>...")
• 15:12, 29 June 2024 Joel2 talk contribs created page File:DysonTorusIllustration03.png
• 15:12, 29 June 2024 Joel2 talk contribs uploaded File:DysonTorusIllustration03.png
• 15:00, 29 June 2024 Joel2 talk contribs created page Apps/DysonPotential (Created page with "__FORCETOC__ <!-- will force the creation of a Table of Contents --> <!-- __NOTOC__ will force TOC off --> =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....")
• 14:57, 29 June 2024 Joel2 talk contribs created page File:FukushimaTorusIllustration02.png