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

### Sunao Ōuchi

Sophia University, Tokyo, Japan

## Abstract

Consider the linear partial differential equation *P*(*z*, ∂_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 → 0 in some sectorial region. We show under some conditions on P(z, ∂_z*) that there exists a solution

*u*(

*z*) which has an asymptotic expansion of the same type as that of

*f*(

*z*).