Existence of Solutions with Asymptotic Expansion of Linear Partial Differential Equations in the Complex Domain

Abstract

Consider the linear partial differential equation $P(z,{\partial }_z)u(z)=f(z)$ in $C^{d+1}$ , where $f(z)$ is not holomorphic on $K=\{z_0=0\}$, but it has an asymptotic expansion with respect to $z_0$ as $z_0\to 0$ in some sectorial region. We show under some conditions on $P(z,{\partial }_z)$ that there exists a solution $u(z)$ which has an asymptotic expansion of the same type as that of $f(z)$.