Integration by differentiation: new proofs, methods and examples

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.

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References in corpus (3)

• New Dirac Delta function based methods with applications to perturbative expansions in quantum field theory
• How to (Path-) Integrate by Differentiating
• More remarkable sinc integrals and sums

Cited by in corpus (4)

• Extension of Stein's lemma derived by using an integration by differentiation technique
• On the concept of a generalized law of refraction: A phenomenological model The Article and the Supporting Informations
• Probabilistic responses of dynamical systems subjected to Gaussian coloured noise excitation. Foundations of a non-Markovian theory
• Towards a more algebraic footing for quantum field theory