Generalized Laurent series for singular solutions of elliptic partial differential equations

Abstract

Introduction. Let L be a linear elliptic differential operator with analytic coefficients in a region R of En. Let L be the adjoint of L. This paper extends the previous work of F. John2 on representation of a solution u of L [u] = 0, where u has a singularity of finite order. A representation is developed here for a solution v of IL [v] = 0, where v has an isolated essential singularity. This representation is a generalization of the Laurent series. Here the summation over the nth powers is replaced by summation over the nth derivatives of a fundamental solution K(x, z), of the operator L. The representation in general is not unique. Uniqueness of a suitably normalized representation is proved for the case in which L is homogeneous with constant coefficients. This gives rise to a theorem which for the three-dimensional Laplace operator reduces to the Maxwell-Sylvester theorem.3