Integration by differentiation: new proofs, methods and examples

TL;DR

This paper rigorously proves new derivative-based methods for solving integrals, motivated by quantum field theory, and demonstrates their effectiveness on Borwein integrals.

Contribution

It provides rigorous proofs for derivative-based integral solving methods and applies them to analyze Borwein integrals.

Findings

Methods are rigorously validated for specific function spaces

Successfully applied to Borwein integrals revealing their behavior

Offers new tools for integral evaluation inspired by quantum physics

Abstract

Recently, new methods were introduced which allow one to solve ordinary integrals by performing only derivatives. These studies were originally motivated by the difficulties of the quantum field theoretic path integral, and correspondingly, the results were derived by heuristic methods. Here, we give rigorous proofs for the methods to hold on fully specified function spaces. We then illustrate the efficacy of the new methods by applying them to the study of the surprising behavior of so-called Borwein integrals.