# 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*).

## Cite this article

Sunao Ōuchi, Existence of Solutions with Asymptotic Expansion of Linear Partial Differential Equations in the Complex Domain. Publ. Res. Inst. Math. Sci. 40 (2004), no. 1, pp. 239–294

DOI 10.2977/PRIMS/1145475972