Exploring Integration by Differentiation

TL;DR

This paper validates and extends the integration by differentiation method, demonstrating its compatibility with classical rules, applying it to various integrals including Ramanujan's, and generalizing it to multivariate and q-integrals.

Contribution

It introduces extensions of the integration by differentiation method to multivariate and q-integrals, and derives a rotationally invariant formulation.

Findings

Validated the method's compatibility with classical integration rules

Applied the method to classical integrals, including Ramanujan's

Extended the method to multivariate and q-integrals

Abstract

This work validates and extends the method of integration by differentiation, initially introduced by A. Kempf et al., and demonstrates its compatibility with classical rules of integration. It provides applications to classical integrals, including one by Ramanujan, and extends the method to the multivariate setting. Volumes of simplexes are computed by acting with indicator functions on elementary kernels, and a rotationally invariant formulation is derived. Finally, the method is extended to Jackson's q-integral.