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

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.