Inhomogeneous Ordinary Differential Equations and Local Cohomologies and Residues

Abstract

Let P = P(z,d/dz) be a linear ordinary differential operator with holomorphic coefficients in a neighbourgood X of the origin 0 in ℂ. Let Pu = f be an inhomogeneous differential equation in the space O {0} or in O {0}, where 0 {0} is the ring of convergent power series and O {0} is the ring of formal power series at the origin 0.