Harmonic function expansion of nearly oblate systems

We show how to develop an expansion of nearly oblate systems in terms of aset of potential-density pairs. A harmonic (multipole) structure is imposed onthe potential set at infinity, and the density can be made everywhere regular.We concentrate on a set whose zeroth order functions describe the perfectoblate spheroid of de Zeeuw (1985). This set is not bi-orthogonal, but it canbe shown to be complete in a weak sense. Poisson's equation can be solvedapproximately by truncating the expansion of the potential in such a set. Asimple example of a potential which is not one of the basis functions isexpanded using the symmetric members of the basis set up to fourth order. Thebasis functions up to first order are reconstructed approximately using 10,000particles to show that this set could be used as part of an $N$-body code.