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H.S. Cohl, J.E. Tohline (LSU)
We show that an exact expression for the Green function in cylindrical coordinates is \begin{eqnarray} {\cal G}( x,x^\prime)= 1/\pi\sqrt{RR^\prime} \summ=-\infty\infty eim(\phi-\phi^\prime) \ Qm-\frac{1}{2}(\chi),\nonumber \end{eqnarray} where \chi\equiv [R2+R\prime^2+(z-z\prime)2]/(2RR\prime), and Qm-\frac{1}{2} is the half-integer order Legendre function of the second kind. This expression is significantly more compact and easier to evaluate numerically than the more familiar cylindrical Green function expression which involves infinite integrals over products of Bessel functions and exponentials. It also contains far fewer terms in its series expansion\ ---\ and is therefore more amenable to accurate evaluation\ ---\ than does the familiar expression for {\cal G}(x, x^\prime) that is given in terms of spherical harmonics. This compact Green function expression is extremely 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.