AbstractLet G be a connected reductive p-adic group and let be its Lie algebra. Let be any G-orbit in . Then the orbital integral corresponding to is an invariant distribution on , and Harish-Chandra proved that its Fourier transform is a locally constant function on the set of regular semisimple elements of . If is a Cartan subalgebra of , and ω is a compact subset of ∩ , we give a formula for (tH) for H ε ω and t ε F× sufficiently large. In the case that is a regular semisimple orbit, the formula is already known by work of Waldspurger. In the case that is a nilpotent orbit, the behavior of at infinity is already known because of its homogeneity properties. The general case combines aspects of these two extreme cases. The formula for at infinity can be used to formulate a “theory of the constant term” for the space of distributions spanned by the Fourier transforms of orbital integrals. It can also be used to show that the Fourier transforms of orbital integrals are “linearly independent at infinity.”