An operational calculus generalization of Ramanujan's master theorem

Abstract

We give a formal extension of Ramanujan's master theorem using operational methods. The resulting identity transforms the computation of an integral of a product on the half-line to the computation of a Laplace transform. Since the identity is purely formal, we show consistency of this operational approach with various standard calculus results, followed by several examples, including a Feynman diagram evaluation, to illustrate the power of the extension. We then briefly discuss the connection between Ramanujan's master theorem and identities of Hardy and Carr before extending the latter identities in the same way we extended Ramanujan's. Finally, we generalize our results, producing additional interesting identities as a corollary.