  We study the asymptotic behaviour of the partial density function associated
to sections of a positive hermitian line bundle that vanish to a particular
order along a fixed divisor $Y$. Assuming the data in question is invariant
under an $S^1$-action (locally around $Y$) we prove that this density function
has a distributional asymptotic expansion that is in fact smooth upon passing
to a suitable real blow-up. Moreover we recover the existence of the "forbidden
region" $R$ on which the density function is exponentially small, and prove
that it has an "error-function" behaviour across the boundary $\partial R$. As
an illustrative application, we use this to study a certain natural function
that can be associated to a divisor in a K\"ahler manifold.
