Editing Appendix/Ramblings/CCGF

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=Compact Cylindrical Green Function (CCGF)=

==Preface by Tohline==
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; 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 most oft-cited research publications and ''certainly'' has proven to be the publication with the most citations from research groups outside of the astrophysics community.  A [[Appendix/Ramblings/CCGF#Sample_Citations_from_Fields_Outside_of_Astronomy|sample of citations from outside the field of astronomy]] is presented, below.  [http://hcohl.sdf.org/bibliography.html Howard Cohl] deserves full credit for the important discovery presented in this paper; I simply tagged along as his ''physics'' doctoral dissertation advisor and harshest skeptic.

<!--
==A Primary Result==
They show, for example, that when expressed in terms of cylindrical coordinates, the axisymmetric potential is,
<div align="center">
<math>
\Phi(R,z) = - \frac{2G}{R^{1/2}} q_0 ,
</math>
</div>
where,
<div align="center">
<math>
q_0 = \int\int (R')^{1/2} \rho(R',z') Q_{-1/2}(\Chi) dR' dz',
</math>
</div>
and the dimensionless argument (the modulus) of the special function, <math>~Q_{-1/2}</math>, is,
<div align="center">
<math>
\Chi \equiv \frac{R^2 + {R'}^2 + (z - z')^2}{2R R'} .
</math>
</div>
Note:  Here we are using <math>~\Chi</math> instead of <math>~\chi</math> (as used by CT99) to represent this dimensionless parameter in order to avoid confusion with our use of <math>~\chi</math>, above.  Next, following the lead of CT99, we note that according to the Abramowitz &amp; Stegun (1965),
<div align="center">
<math>Q_{-1/2}(\Chi) = \mu K(\mu) \, ,</math>
</div>
where, the function <math>~K(\mu)</math> is the complete elliptical integral of the first kind and, for our particular problem,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~2(1+\Chi)^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\biggl[ 1+\frac{R^2 + {R'}^2 + (z - z')^2}{2R R'} \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{4R R'}{(R + {R'})^2 + (z - z')^2} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, we can write,
<div align="center">
<math>
q_0 = \int\int (R')^{1/2} \rho(R',z') \mu K(\mu) dR' dz' \, .
</math>
</div>
-->

==Sample Citations from Fields Outside of Astronomy==

===Journal of Quantitative Spectroscopy and Radiative Transfer===
<ul>
<li>
<font color="red">[2019]</font> ''Electrostatic T-matrix for a torus on bases of toroidal and spherical harmonics'', by M. Majic, [https://ui.adsabs.harvard.edu/abs/2019JQSRT.235..287M/abstract JQSRT, Volume 235, pp. 287-299] 
</li>
</ul>

===Journal of Computational Physics===
<ul>
<li>
<font color="red">[2018]</font> ''An integral equation-based numerical solver for Taylor states in toroidal geometries'', by M. O'Neil &amp; A. J. Cerfon, [https://www.sciencedirect.com/science/article/pii/S0021999118300147 J. Comp. Phys., Volume 359, pp. 263-282] 
</li>
<li>
<font color="red">[2016]</font> ''Determination of normalized electric eigenfields in microwave cavities with sharp edges'', by J. Helsing &amp; A. Karlsson, [https://doi.org/10.1016/j.jcp.2015.09.054 J. Comp. Phys., Volume 304, pp. 465-486] 
</li>
<li>
<font color="red">[2014]</font> ''An explicit kernel-split panel-based Nystr&ouml;m scheme for integral equations on axially symmetric surfaces'', by J. Helsing &amp; A. Karlsson, [https://www.sciencedirect.com/science/article/pii/S0021999114003295 J. Comp. Phys., Volume 272, pp. 686-703] 
</li>
</ul>

===Journal of Physics A: &nbsp; Mathematical and Theoretical===
<ul>
<li>
<font color="red">[2007]</font> ''A method for studying electron-density-based dynamics of many-electron systems in scaled cylindrical coordinates'', by A. Poddar &amp; B. M. Deb, [http://iopscience.iop.org/article/10.1088/1751-8113/40/22/015/meta Journal of Physics A:  Mathematical and Theoretical, Volume 40, pp. Number 22] 
</li>
</ul>

===Physical Review B===
<ul>
<li>
<font color="red">[2014]</font> ''Spin and impurity effects on flux-periodic oscillations in core-shell nanowires'', by T. O. Rosdahl, A. Manolescu, &amp; V. Gudmundsson, [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.035421 Phys. Rev. B 90, 035421] &#8212; The key reference to CCGF appears in the paragraph associated with their equation (30); the authors state that numerical evaluation of the relevant set of Legendre functions was carried out using a code provided in [https://www.sciencedirect.com/science/article/pii/S0010465599004282?via%3Dihub J. Segura &amp; A. Gil, Comput. Phys. Commun. 124, 104, (2000)]
</li>
<li>
<font color="red">[2011]</font> ''Transverse domain wall propagation in modulated cylindrical nanostructure and possible geometric control'', by S. Allende, &amp; R. Arias, [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.83.174452 Phys. Rev. B 83, 174452] 
</li>
<li>
<font color="red">[2005]</font> ''Ground-state densities and pair correlation functions in parabolic quantum dots'', by M. Gattobigio, P. Capuzzi, M. Polini, R. Asgari, &amp; M. P. Tosi, [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.72.045306 Phys. Rev. B 72, 045306] 
</li>
</ul>

===Physical Review C===
<ul>
<li>
<font color="red">[2010]</font> ''Linear response of light deformed nuclei investigated by self-consistent quasiparticle random-phase approximation'', by C. Losa, A. Pastore, T. D&oslash;ssing, E. Vigezzi, &amp; R. A. Broglia, [https://journals.aps.org/prc/abstract/10.1103/PhysRevC.81.064307 Phys. Rev. C 81, 064307] 
</li>
</ul>

===Physical Review D===
<ul>
<li>
<font color="red">[2019]</font> ''Closed form expressions for gravitational multipole moments of elementary solids'', by J. Stirling &amp; S. Schlamminger, [https://ui.adsabs.harvard.edu/abs/2019PhRvD.100l4053S/abstract Phys. Rev. D, Vol. 100, Issue 12, 124053] 
</li>
</ul>

===Journal of the Mechanics and Physics of Solids===
<ul>
<li>
<font color="red">[2016]</font> ''Cyclic density functional theory: &nbsp; A route to the first principles simulation of bending in nanostructures'', by A. S. Banerjee &amp; P. Suryanarayana, [https://www.sciencedirect.com/science/article/pii/S0022509616303684 Journal of the Mechanics and Physics of Solids 96, pp. 605-631] 
</li>
</ul>

===Journal of Applied Physics===
<ul>
<li>
<font color="red">[2013]</font> ''Reflectance modulation by free-carrier exciton screening in semiconducting nanotubes'', by F. Pinto, [https://doi.org/10.1063/1.4812495 Journal of Applied Physics, Volume 114, 024310] 
</li>
<li>
<font color="red">[2010]</font> ''Magnetoexcitons in electron-hole bilayer nanotubes made of rolled-up type-II band aligned quantum wells'', by M. Bagheri, [http://aip.scitation.org/doi/abs/10.1063/1.3428436 Journal of Applied Physics, Volume 107, 114305] 
</li>
</ul>

===IEEE Transactions on Microwave Theory and Techniques===
<ul>
<li>
<font color="red">[2015]</font> ''Determination of Normalized Magnetic Eigenfields in Microwave Cavities'', by J. Helsing &amp; A. Karlsson, [http://ieeexplore.ieee.org/abstract/document/7061541/ IEEE Transactions on Microwave Theory and Techniques, Volume 63, Issue 5] &#8212; Relevant to "&hellip; the development of medical instrumentation &hellip;"
</li>
</ul>

===IEEE Transactions on Magnetics===
<ul>
<li>
<font color="red">[2015]</font> ''Field distribution around surface cracks in metallic cylindrical structures excited by high-frequency current-carrying coils of arbitrary shape'', by A. Akbari-Khezri, S. H. H. Sadeghi, &amp; R. Moini, [http://ieeexplore.ieee.org/abstract/document/6880810/ IEEE Transactions on Magnetics, Volume 51, Issue 2] 
</li>
<li>
<font color="red">[2013]</font> ''Optimal Configuration for Electromagnets and Coils in Magnetic Actuators'', by S. Afshar, M. B. Khamesee, &amp; A. Khajepour, [http://ieeexplore.ieee.org/abstract/document/6376203/ IEEE Transactions on Magnetics, Volume 49, Issue 4] &#8212; Relevant to "&hellip; the development of medical instrumentation &hellip;"
</li>
<li>
<font color="red">[2007]</font> ''Computation of the three-dimensional magnetic field from solid permanent-magnet bipolar cylinders by employing toroidal harmonics'', by J. P. Selvaggi, S. Salon, &amp; O.-Mun Kwon, [http://ieeexplore.ieee.org/abstract/document/4303214/authors IEEE Transactions on Magnetics, Volume 43, Issue 10] &#8212; Relevant to "&hellip; the development of medical instrumentation &hellip;"
</li>
</ul>

===IEEE Transactions on Antennas and Propagation===
<ul>
<li>
<sup>&dagger;</sup><font color="red">[2005]</font> ''New exact solution procedure for the near fields of the general thin circular loop antenna'', by J. T. Conway, [http://ieeexplore.ieee.org/abstract/document/1377629/  IEEE Transactions on Antennas and Propagation, Volume 53, Issue 1] 
</li>
</ul>

===New Journal of Physics===
<ul>
<li>
<font color="red">[2008]</font> ''Calculation of electrostatic fields using quasi-Green's functions:  application to the hybrid Penning trap'', by J. Veru&uacute;, S. Kreim, K. Blaum, H. Kracke, W. Quint, S. Ulmer, &amp; J. Walz, [http://iopscience.iop.org/article/10.1088/1367-2630/10/10/103009/meta New Journal of Physics, Volume 10, October] 
</li>
</ul>

===Journal of Molecular Physics===
<ul>
<li>
<font color="red">[2005]</font> ''Scaling in complex systems:  analytical theory of charged pores'', by A. Enriquez &amp; L. Blum, [http://www.tandfonline.com/doi/abs/10.1080/00268970500221941 J. Molecular Physics, Volume 103, pp. 3201-3208] 
</li>
</ul>

===Plasma Physics &hellip;===
====Physics of Plasmas====
<ul>
<li>
<font color="red">[2019]</font> ''Expressions for perturbed vacuum potential energy for 3D linear MHD stability'', by T. Weyens, [https://ui.adsabs.harvard.edu/abs/2019PhPl...26d2507W/abstract Physics of Plasmas, 26, Issue 4, 042507] 
</li>
<li>
<font color="red">[2018]</font> ''Reaction of the toroidal resistive wall on the magnetic field variations in tokamak-like systems'', by V. D. Pustovitov, [https://ui.adsabs.harvard.edu/abs/2018PhPl...25f2510P/abstract Physics of Plasmas, 25, 062510] 
</li>
<li>
<font color="red">[2008]</font> ''General formulation of the resistive wall mode coupling equations'', by V. D. Pustovitov, [http://aip.scitation.org/doi/abs/10.1063/1.2943711 Physics of Plasmas, 15, 072501] &#8212; Relevant to "&hellip; toroidal plasmas &hellip;"
</li>
</ul>

====Plasma Physics and Controlled Fusion====
<ul>
<li>
<font color="red">[2008]</font> ''Decoupling in the problem of tokamak plasma response to asymmetric magnetic perturbations'', by V. D. Pustovitov, [http://iopscience.iop.org/article/10.1088/0741-3335/50/10/105001/meta Plasma Physics and Controlled Fusion, Volume 50, Number 10] 
</li>
</ul>

====Plasma Physics Reports====
<ul>
<li>
<font color="red">[2019]</font> ''Extension of Shafranov's Equilibrium Theory to the Description of Current Quenches Affected by Resistive Wall Dissipation in Tokamaks'', by V. D. Pustovitov, [https://ui.adsabs.harvard.edu/abs/2019PlPhR..45.1114P/abstract Plasma Physics Reports, Volume 45, Issue 12, p. 1114 - 1127] 
</li>
</ul>

==Selected Citations from Astrophysicists==
<ol>
<li>
(2019) ''A Fast Poisson Solver of Second-order Accuracy for Isolated Systems in Three-dimensional Cartesian and Cylindrical Coordinates'', by S. Moon, W.-T. Kim &amp; E. C. Ostriker, [https://ui.adsabs.harvard.edu/abs/2019ApJS..241...24M/abstract ApJS, Volume 241, Issue 2, 24]
</li>
<li>
(2019) ''AGAMA: action-based galaxy modelling architecture'', by E. Vasiliev, [https://ui.adsabs.harvard.edu/abs/2019MNRAS.482.1525V/abstract MNRAS, Volume 482, Issue 2, p. 1525 - 1544]
</li>
<li>
(2016) ''Equilibrium sequences and gravitational instability of rotating isothermal rings'', by W.-T. Kim &amp; S. Moon, [http://iopscience.iop.org/article/10.3847/0004-637X/829/1/45/meta ApJ, Volume 829, Number 1]
</li>
<li>
(2016) ''Zonal toroidal harmonic expansions of external gravitational fields for ring-like objects'', by T. Fukushima, [http://iopscience.iop.org/article/10.3847/0004-6256/152/2/35/meta The Astronomical Journal, Volume 152, Number 2]
</li>
<li>
<sup>&dagger;</sup>(2016) ''Vector potentials for the gravitational interaction of extended bodies and laminas with analytical solutions for two disks'', by J. T. Conway, [https://link.springer.com/article/10.1007/s10569-016-9679-y Celestial Mechanics and Dynamical Astronomy, Volume 125, Issue 2, pp. 161-194]
</li>
<li>
(2015) ''Applying Schwarzschild's orbit superposition method to barred or non-barred disc galaxies'', by E. Vasiliev &amp; E. Athanassoula, [https://academic.oup.com/mnras/article/367/3/1297/1042268 MNRAS, Volume 450, Issue 3]
</li>
<li>
(2012) ''A substitute for the singular Green kernel in the Newtonian potential of celestial bodies'', by J.-M. Our&eacute; &amp; A. Dieckmann, [https://www.aanda.org/articles/aa/pdf/forth/aa18443-11.pdf Astronomy &amp; Astrophysics, Volume 541, A130]
</li>
<li>
(2007) ''The Newtonian force experienced by a point mass near a finite cylindrical source'', by J. P. Selvaggi, Sheppard Salon &amp; M. V. K. Chari, [http://iopscience.iop.org/article/10.1088/0264-9381/25/1/015013/meta Classical and Quantum Gravity, Volume 25, Number 1]
</li>
<li>
(2006) ''Self-consistent response of a galactic disc to vertical perturbations'', by K. Saha &amp; C. J. Jog, [https://academic.oup.com/mnras/article-abstract/450/3/2842/1068852 MNRAS, Volume 367, Issue 3]
</li>
<li>
(2005) ''Accurate numerical potential and field in razor-thin, axisymmetric disks'', by J.-M. Our&eacute;, [http://iopscience.iop.org/article/10.1086/428769/meta ApJ, Volume 624, Number 1]
</li>
<li>
(2004) ''Evolution of self-gravitating magnetized disks.  I. Axisymmetric simulations'', by S. Forming, S. A. Balbus, &amp; J.-P. De Villers, [http://iopscience.iop.org/article/10.1086/424828/meta ApJ, Volume 616, Number 1]
</li>
</ol>

=See Also=
* <sup>&dagger;</sup>[https://www.uia.no/kk/profil/johntc John Thomas Conway] has authored many articles &#8212; appearing in journals covering a wide range of disciplines &#8212; whose research topics overlap, if not incorporate, the [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)] work. 

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