Application of the Efros Theorem to the Function Represented by the Inverse Laplace Transform of s−μ exp(−sν)

Alexander Apelblat,Francesco Mainardi

doi:10.3390/sym13020354

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−μexp(−sν) with μ≥0 and 0<ν<1 are presented.

Highlights

Inversions of the Laplace transforms of exponential functionsInverse Laplace Transform of s−μ exp(−sν )
The logarithmic functions in the Laplace transforms permit to express the integrals of the Volterra functions with inverse Laplace transform of exponential function in terms of convolution integrals
By applying the Efros theorem in the form established by Wlodarski it was possible to derive a number of infinite integrals, finite integrals and integral identities with the function which represent the Laplace inverse transform of s−μ exp(−sν ) with 0 < ν < 1 and 0 ≤ μ < 1

Summary

Introduction

Inversions of the Laplace transforms of exponential functionsInverse Laplace Transform of s−μ exp(−sν ). Laplace transform in terms of integral representations [7] 2. Integral Representations of the Inverse Laplace Transform of s−μ exp(−sν )

Results

Conclusion