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In this article we develop a method of finding the static axisymmetricspace-time corresponding to any given set of multipole moments. In addition toan implicit algebraic form for the general solution, we also give a powerseries expression for all finite sets of multipole moments. As conjectured byGeroch we prove in the special case of axisymmetry, that there is a staticspace-time for any given set of multipole moments subject to a (specified)convergence criterion. We also use this method to confirm a conjecture ofHernandez-Pastora and Martin concerning the monopole-quadropole solution.Comment: 14 page

Static axisymmetric spacetimes with prescribed multipole moments