A Compact Cylindrical Green's Function Expansion for the Solution of Potential Problems
Abstract
We show that an exact expression for the Green's function in cylindrical coordinates is1/(xx^{'})=1/(πsqrt(RR^{'}))Σm=∞∞e^{im(φφ')}Q_{m1/2}(χ) , where χ≡[R^{2}+R^{'2}+(zz^{'})^{2}]/(2RR^{'}), and Q_{m1/2} is the halfinteger degree Legendre function of the second kind. This expression is significantly more compact and easier to evaluate numerically than the more familiar cylindrical Green's function expression, which involves infinite integrals over products of Bessel functions and exponentials. It also contains far fewer terms in its series expansionand is therefore more amenable to accurate evaluationthan does the familiar expression for xx^{'}^{1} that is given in terms of spherical harmonics. This compact Green's function expression is well suited for the solution of potential problems in a wide variety of astrophysical contexts because it adapts readily to extremely flattened (or extremely elongated), isolated mass distributions.
 Publication:

The Astrophysical Journal
 Pub Date:
 December 1999
 DOI:
 10.1086/308062
 Bibcode:
 1999ApJ...527...86C
 Keywords:

 GALAXIES: FORMATION;
 GALAXIES: STRUCTURE;
 METHODS: ANALYTICAL;
 STARS: FORMATION;
 Galaxies: Formation;
 Galaxies: Structure;
 Methods: Analytical;
 Stars: Formation