Let $B$ be a compact subset in $\mathbb R^n$ and $J:\mathbb R^n\to\mathbb R$ with $J^2$ being real analytic and $J\neq0$ almost everywhere. Then $\int_BJ(x)^{-2\delta}dx$ is convergent for $\delta$ close to zero by the Weierstrass preparation theorem.
I can't see how the theorem is applied here. Could anyone fill in more details?