The fundamental theorem of calculus in differential rings
Clemens G. Raab, Georg Regensburger
TL;DR
The paper develops an algebraic framework for the fundamental theorem of calculus by introducing generalized integro-differential rings with a nonmultiplicative evaluation. It then constructs the ring of integro-differential operators (IDO) and a complete rewrite system to obtain normal forms, enabling purely algebraic proofs of classical results such as variation of constants and Taylor-type formulas, even for systems with matrix coefficients. A generalized shuffle theory for nested integrals is developed, with evaluation terms accounting for singularities, and the approach extends to linear systems and initial-value problems. Overall, the work unifies differentiation, integration, and evaluation inside an algebraic structure, providing tools for symbolic manipulation and boundary-value analysis beyond traditional analytic constraints.
Abstract
In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such integro-differential rings and discuss many examples. We also construct corresponding integro-differential operators and provide normal forms via rewrite rules. They are then used to derive several identities and properties in a purely algebraic way, generalizing well-known results from analysis. In identities like shuffle relations for nested integrals and the Taylor formula, additional terms are obtained that take singularities into account. Another focus lies on treating basics of linear ODEs in this framework of integro-differential operators. These operators can have matrix coefficients, which allow to treat systems of arbitrary size in a unified way. In the appendix, using tensor reduction systems, we give the technical details of normal forms and prove them for operators including other functionals besides evaluation.