Representation of Solutions with Non-Discrete Singularities

Abstract

Let K be a compact subset of the complex space ℂⁿ. Given a family ℱ of functions analytic on K, we would like to know when there exists a single neighborhood of K to which every function in ℱ can be extended. Elementary consideration provides a simple necessary condition: there exists a positive number r such that for each point y in K and each function u in ℱ, the radius of convergence of the Taylor series of u at y is at least r. We call a compact set K regular relative to ℱ if this necessary condition is also sufficient. We are especially interested in the case where K lies in the “real” subspace ℝⁿ of ℂⁿ and ℱ is a family of solutions of an elliptic system of differential equations with real analytic coefficients. The principal result of this chapter is that if K is regular relative to the family of solutions of the transposed system on K, then the solutions of the original system away from K admit appropriate expansions in solutions with pole-type singularities on K.