  We describe a technique to analytically compute the multipole moments of a
charge distribution confined to a planar triangle, which may be useful in
solving the Laplace equation using the fast multipole boundary element method
(FMBEM) and for charged particle tracking. This algorithm proceeds by
performing the necessary integration recursively within a specific coordinate
system, and then transforming the moments into the global coordinate system
through the application of rotation and translation operators. This method has
been implemented and found use in conjunction with a simple piecewise constant
collocation scheme, but is generalizable to non-uniform charge densities. When
applied to low aspect ratio ($\leq 100$) triangles and expansions with degree
up to 32, it is accurate and efficient compared to simple two-dimensional
Gauss-Legendre quadrature.
