The Cauchy principal value and the Hadamard finite part integral as values of absolutely convergent integrals

Abstract

The divergent integral ∫abf(x)(x−x0)−n−1dx, for −∞ < a < x0 < b < ∞ and n = 0, 1, 2, …, is assigned, under certain conditions, the value equal to the simple average of the contour integrals ∫C±f(z)(z − x0)−n−1dz, where C+ (C−) is a path that starts from a and ends at b and which passes above (below) the pole at x0. It is shown that this value, which we refer to as the analytic principal value, is equal to the Cauchy principal value for n = 0 and to the Hadamard finite-part of the divergent integral for positive integer n. This implies that, where the conditions apply, the Cauchy principal value and the Hadamard finite-part integral are in fact values of absolutely convergent integrals. Moreover, it leads to the replacement of the boundary values in the Sokhotski-Plemelj-Fox theorem with integrals along some arbitrary paths. The utility of the analytic principal value in the numerical, analytical, and asymptotic evaluations of the principal value and the finite-part integral is discussed and demonstrated.