Inverse Laplace transform based on Widder's method for Tsallis exponential

TL;DR

This paper introduces a generalized Laplace transform based on Tsallis' q-exponential, along with a novel inverse transform using Post-Widder's method, facilitating computations in statistical mechanics.

Contribution

It develops a new q-generalized inverse Laplace transform using Post-Widder's method, avoiding complex contour integrals and applying it to physical models.

Findings

Derived a q-generalized inverse Laplace transform

Applied the method to compute density of states for physical systems

Validated the approach with elementary functions and physical models

Abstract

A generalization of the Laplace transform based on the generalized Tsallis $q$-exponential is given in the present work for a new type of kernel. We also define the inverse transform for this generalized transform based on the complex integration method. We prove identities corresponding to the Laplace transform and inverse transform like the $q$-convolution theorem, the action of generalized derivative and generalized integration on the Laplace transform. We then derive a $q$-generalization of the inverse Laplace transform based on the Post-Widder's method which bypasses the necessity for a complex contour integration. We demonstrate the usefulness of this in computing the Laplace and inverse Laplace transform of some elementary functions. Finally we use the Post-Widder's method based inverse Laplace transform to compute the density of states from the partition function for the case of a generalized classical ideal gas and linear harmonic oscillator in $D$-dimensions.