Suppose $f:D \to \mathbb{C}$ is holomorphic in a domain $D \subset \mathbb{C}^n$, all of its partials extend continuously to $\bar{D}$ and $f$ is meromorphic in a neighborhood of $\bar{D}$. Suppose $0 \in \partial D$. In a small neighborhood of 0, why is
$$ f(z) = q(z) + \frac{\sum_{j=0}^{m-1}b_j(z')z_n^j}{z_n^m + \sum_{j=0}^{m-1}a_j(z')z_n^j} $$
where $q$ is holomorphic, and the $a_j$'s and $b_j$'s are holomorphic functions of $z' = (z_1, ... z_{n-1})$ which vanish at $0' \in C^{n-1}$.
This result reminds me of the Weierstrass Preparation Theorem because the numerator and denominator have coefficients in $z'$ and vanish at 0'.