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

In this paper, we consider the inverse Laplace transform of the Volterra mu-function (the Bromwich integral) and evaluate it with respect to the Hankel, Schlafli and Bourguet complex contours. In this sense, we establish the generalized Ramanujan's integral representations of the Volterra mu-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 mu-function.

Automated summary

In this paper, the inverse Laplace transform of the Volterra mu-function and its evaluation using different complex contours are considered. The generalized Ramanujan's integral representations for the Volterra mu-function with general variations of the parameters are established. The asymptotic analysis of this function with large parameters using the steepest descent method is also discussed. Furthermore, it is shown that the solution of the Volterra integral equation with a differentiated-order fractional integral operator is the Volterra mu-function.