Static axisymmetric spacetimes with prescribed multipole moments

In this paper we develop a method of finding the static axisymmetric spacetime corresponding to any given set of multipole moments. In addition to an implicit algebraic form for the general solution, we also give a power series expression for all finite sets of multipole moments. As conjectured by Geroch we prove in the special case of axisymmetry, that there is a static spacetime for any given set of multipole moments subject to a (specified) convergence criterion. We also use this method to confirm a conjecture of Hemdndez-Pastora and Martin concerning the monopole-quadrupole solution.