Generalization of the Ramanujan's integrals for the Volterra μ -functions via complex contours: representations and approximations

AbstractIn this paper, we consider the inverse Laplace transform of the Volterra μ-function (the Bromwich integral) and evaluate it with respect to the Hankel, Schläfli and Bourguet complex contours. In this sense, we establish the generalized Ramanujan's integral representations of the Volterra μ-function for general variations of the parameters. We also discuss the asymptotic analysis of this function with large parameters using the steepest descent method. Further, we show that the solution of Volterra integral equation with differentiated-order fractional integral operator is the Volterra μ-function.Keywords: Volterra μ-functionRamanujan's integralLaplace transformMathematics Subject Classifications: 41A6044A1045D05 AcknowledgmentsThe authors would like to acknowledge the reviewer for the helpful, constructive and encouraging comments.Disclosure statementNo potential conflict of interest was reported by the author(s).