Integration by divergent integrals: Calculus of divergent integrals in term by term integration

TL;DR

This paper develops a calculus for divergent integrals in term-by-term integration, showing that assigning arbitrary values to divergences can recover missing terms and unify different interpretations.

Contribution

It formulates a unified calculus for divergent integrals, allowing arbitrary value assignments that automatically include missing terms in term-by-term integration.

Findings

Divergent integrals can be assigned arbitrary values within a unified calculus.

Missing terms in divergent integral evaluations emerge from interpretation choices.

The calculus generalizes previous finite part integral methods to broader divergent integral interpretations.

Abstract

It is known in the case of the Stieltjes transform that evaluating the integral by expanding the kernel of transformation followed by term by term integration leads to an infinite series of divergent integrals. Moreover, it is known that merely assigning values to the divergent integrals leads to missing terms. This problem of missing terms was earlier resolved in the complex plane, which required the divergent integrals to assume values equal to their finite parts under a set of rules or calculus governing the use of the finite part integral [E.A. Galapon, {\it Proc. Roy. Soc. A} {\bf 473}, 20160567 (2017)]. However, divergent integrals assume a spectrum of values arising from the different ways of removing the divergences in the integration. Here we show that the other interpretations of divergent integrals follow the same set of rules as those of the finite part integral, and then formulate the calculus of divergent integrals arising from term-by-term integration. We find that the calculus does not prescribe a specific interpretation to the divergent integrals but allows us to assign arbitrary values to them, with the missing terms automatically emerging from the chosen interpretation of the divergent integrals itself.