Annihilators of Laurent coefficients of the complex power for normal crossing singularity

TL;DR

This paper identifies differential operators that annihilate the Laurent coefficients of the complex power distribution of a real analytic function with normal crossing singularities, advancing understanding of their structure.

Contribution

It explicitly determines the annihilators of Laurent coefficients for complex powers of functions with normal crossing singularities, a novel result in distribution theory.

Findings

Explicit annihilators for Laurent coefficients are derived.

Results apply to functions with normal crossing singularities.

Enhances understanding of complex power distributions near singularities.

Abstract

Let $f$ be a real-valued real analytic function defined on an open set of $\mathbb{R}^n$. Then the complex power $f_+^\lambda$ is defined as a distribution with a holomorphic parameter $\lambda$. We determine the annihilator (in the ring of differential operators) of each coefficient of the principal part of the Laurent expansion of $f_+^\lambda$ about $\lambda=-1$ in case $f=0$ has a normal crossing singularity.