Exploring Integration by Differentiation
R. D. George, C. Vignat
TL;DR
The paper presents the integration-by-differentiation framework, recasting Laplace-type integrals as the action of pseudo-differential operators on simple kernels and extending the method to multivariate and q-calculus settings. It validates compatibility with standard rules, demonstrates concrete one- and multi-dimensional applications (including trigonometric and Ramanujan-type integrals), and shows how indicator functions and rotational invariance simplify volume and transform computations. By tensorizing the method across variables and leveraging Euler-type representations, it derives closed forms and univariate reductions for complex multidimensional integrals, including simplex volumes. The work also extends the approach to Jackson's q-integral, providing explicit q-analogues and highlighting potential for further operator-based kernel constructions. Overall, it positions integration by differentiation as a powerful, versatile symbolic technique with connections to Laplace/Fourier analysis, indicator-function methods, and q-calculus for a broad class of integrals and volumes.
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.