**AAS Meeting #193 - Austin, Texas, January 1999**

*Session 120. Computational Techniques, Catalogs and Literature*

Oral, Saturday, January 9, 1999, 2:00-3:30pm, Room 9 (C)
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## [120.02] An Extraordinarily Compact Cylindrical Green Function Expansion for the Solution of Potential Problems

*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}
\sum_{m=-\infty}^{\infty} e^{im(\phi-\phi^\prime)} \
Q_{m-\frac{1}{2}}(\chi),\nonumber \end{eqnarray} where
\chi\equiv
[R^{2}+R^{\prime^2}+(z-z^{\}prime)^{2}]/(2RR^{\prime}), and
Q_{m-\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.

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