  Let $M$ be a smooth manifold and $G$ a compact connected Lie group acting on
$M$ by isometries. In this paper, we study the equivariant cohomology of ${\bf
X}=T^\ast M$, and relate it to the cohomology of the Marsden-Weinstein reduced
space via certain residue formulae. In case that $\bf X$ is a compact
symplectic manifold with a Hamiltonian $G$-action, similar residue formulae
were derived by Jeffrey, Kirwan et al.
