Ramanujan’s Master Theorem

Abstract

S. Ramanujan introduced a technique, known as Ramanujan's Master Theorem, which provides an analytic expression for the Mellin transform of a function. The main identity of this theorem involves the extrapolation of the sequence of coefficients of the integrand, defined originally as a function on N to C. The history and proof of this result are reviewed. Applications to the evaluation of a variety of definite integrals is presented.

Three-Step and Four-Step Random Walk Integrals

“…♦ Remark 1. Here, we demonstrate how Ramanujan's "master theorem" may be applied to find the Bessel integral representation (11) in a natural way; this and more applications of Ramanujan's master theorem will appear in [RMT10]. For an alternative proof see [Bro09].…”

The Umbral operator and the integration involving generalized Bessel-type functions

“…This identity has been proved rigorously in [1,2]. A simpler, albeit heuristic, proof can be achieved by exploiting umbral method [18] can be found in [4,9].…”

Testing a dissipative kinetic k-essence model

“…This work can be improved in many ways. On the one hand, we can attempt an alternative way to obtain analytical solutions, a possibility we are already studying using a novel technique proposed to solve complex differential equations [93][94][95][96][97][98]. In this case, we have the possibility to consider λ a free parameter, enhancing the parameter space to find a best fit with the data.…”