The fundamental theorem of calculus in differential rings

TL;DR

This paper explores the algebraic structure of calculus through integro-differential rings, extending classical results to functions with singularities and developing a framework for operators with matrix coefficients.

Contribution

It introduces a generalized algebraic framework for calculus, including evaluation at singularities, and constructs normal forms for integro-differential operators with applications to linear ODEs.

Findings

Generalized evaluation for singular functions

Normal forms for integro-differential operators

Unified treatment of systems of linear ODEs

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.