Application of the Efros theorem to the function represented by the inverse Laplace transform of $s^{-\mu}\,\exp(-s^\nu)$

TL;DR

This paper applies the Efros theorem and operational calculus to derive new integral identities and properties for the inverse Laplace transform of a specific function involving parameters, with applications to special functions.

Contribution

It introduces novel integral identities and properties for the inverse Laplace transform of s^{-d} e^{-s^ u} using Efros theorem and operational calculus.

Findings

Derived new integral identities involving Mittag-Leffler and Volterra functions.

Established properties of the inverse Laplace transform for specific parameter ranges.

Connected integral identities to elementary and special functions.

Abstract

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag-Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag-Leffler functions and the Volterra functions. Some properties of the inverse Laplace transform of $s^{-\mu} \exp(-s^\nu)$ with $\mu \ge0$ and $0<\nu<1$ are presented