The problem of missing terms in term by term integration involving divergent integrals

TL;DR

This paper addresses the issue of missing terms in term-by-term integration involving divergent integrals, resolving it by complex plane lifting under certain analyticity conditions, and identifying the source of missing terms as singularities.

Contribution

It introduces a method to recover missing terms in divergent integrals by lifting integration into the complex plane, clarifying their origin from singularities.

Findings

Missing terms originate from singularities of the complex integrand.

Lifting integration into the complex plane resolves missing terms.

Divergent integrals are interpreted as finite part integrals.

Abstract

Term by term integration may lead to divergent integrals, and naive evaluation of them by means of, say, analytic continuation or by regularization or by the finite part integral may lead to missing terms. Here, under certain analyticity condition, the problem of missing terms for the incomplete Stieltjes transform, $\int_0^a f(x) (\omega+x)^{-1} \mathrm{d}x$, and the Stieltjes transform itself, $\int_0^{\infty} f(x) (\omega+x)^{-1} \mathrm{d}x$, is resolved by lifting the integration in the complex plane. It is shown that the missing terms arise from the singularities of the complex valued function $f(z) (\omega + z)^{-1}$, with the divergent integrals arising from term by term integration interpreted as finite part integrals.