A compact cylindrical Green's function expansion for the solution of potential problems

We show that an exact expression for the Green's function in cylindrical co ordinates is [GRAPHICS] where chi = [R-2 + R'(2) + (z - z')(2)]/(2RR'), and Q(m-1/2) is the half-in teger degree Legendre function of the second kind. This expression is signi ficantly more compact and easier to evaluate numerically than the more fami liar cylindrical Green's function expression, which involves infinite integ rals over products of Bessel functions and exponentials. It also contains f ar fewer terms in its series expansion-and is therefore more amenable to ac curate evaluation-than does the familiar expression for \x - x'\(-1) that i s given in terms of spherical harmonics. This compact Green's function expr ession is well suited for the solution of potential problems in a wide vari ety of astrophysical contexts because it adapts readily to extremely flatte ned (or extremely elongated), isolated mass distributions.