Many-valuedness of the finite part integral

Abstract

We considered the finite parts of divergent integrals of the form ∫_a^b g(x) (x-x₀)^-n dx for a<x₀<b. It is shown that the finite part takes on a broad range of values, depending on how the contribution of the offending singularity was removed. A conditionally convergent logarithmic term appeared for cases in which the two radii of the contours used possessed a linear dependence towards each other, proving that the finite part could take on many values. The multi-valued finite parts were finally expressed as contour integral representations in the complex plane.