Asymptotic Expansion of Singular Solutions and the Characteristic Polygon of Linear Partial Differential Equations in the Complex Domain

Let P(z, d) be a linear partial differential operator with holomorphic coefficients in a neighborhood Q of z = 0 in C. Consider the equation P(z,d)u(z) = f ( z ) , where u(z) admits singularities on the surface K = {ZQ = 0} and f ( z ) has an asymptotic expansion of Gevrey type with respect to ZQ as ~Q —>• 0. We study the possibility of asymptotic expansion of u ( z ) . We define the characteristic polygon of P(z, d) with respect to K and characteristic indices. We discuss the behavior of u(z) in a neighborhood of K, by using these notions. The main result is a generalization of that in [6].