The Cauchy Principal Value and the finite part integral as values of absolutely convergent integrals

TL;DR

This paper introduces the Analytic Principal Value, showing it unifies the Cauchy principal value and finite-part integral as absolutely convergent integrals, enhancing their numerical and analytical evaluation methods.

Contribution

It establishes the Analytic Principal Value as a unifying concept for divergent integrals, replacing boundary values with arbitrary path integrals, and demonstrates its utility in various evaluations.

Findings

Analytic Principal Value equals Cauchy principal value for n=0.

Analytic Principal Value equals finite-part integral for positive n.

Facilitates numerical and asymptotic evaluation of divergent integrals.

Abstract

The divergent integral $\int_a^b f(x)(x-x_0)^{-n-1}\mathrm{d}x$, for $-\infty<a<x_0<b<\infty$ and $n=0, 1, 2, \dots$, is assigned, under certain conditions, the value equal to the simple average of the contour integrals $\int_{C^{\pm}} f(z)(z-x_0)^{-n-1}\mathrm{d}z$, where $C^+$ ($C^-$) is a path that starts from $a$ and ends at $b$, and which passes above (below) the pole at $x_0$. 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 finite-part of the divergent integral for positive integer $n$. This implies that, where the conditions apply, the Cauchy principal value and the 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 evaluation of the Cauchy principal value and the finite-part integral is discussed and demonstrated.