Real Principal Value Integrals

 We study principal value integrals of multi-valued differential forms on compact spaces, as introduced by Langlands. Using resolutions of singularities we extend Langlands definition to the case in which the differential form may have no normal crossings. We show by an example that for non-compact spaces the principal value integral associated to a compactification may depend on the compactification. Principal value integrals appear as residues of poles of distributions and as coefficients of asymptotic expansions of oscillating integrals and fibre integrals. (Received 11 November 1999)