Analytical calculation of the inverse nabla Laplace transform

TL;DR

This paper develops and compares two classical methods, residual calculation and partial fraction, for analytically computing the inverse nabla Laplace transform, including extensions to fractional order cases.

Contribution

It introduces new formulas and a comprehensive table for inverse nabla Laplace transform, enhancing analytical capabilities for causal sequences and fractional order systems.

Findings

Effective inverse transform formulas are derived for poles inside and outside the contour.

A transform pair table for common functions is established.

Methods are validated with illustrative examples and extended to fractional cases.

Abstract

The inversion of nabla Laplace transform, corresponding to a causal sequence, is considered. Two classical methods, i.e., residual calculation method and partial fraction method are developed to perform the inverse nabla Laplace transform. For the first method, two alternative formulae are proposed when adopting the poles inside or outside of the contour, respectively. For the second method, a table on the transform pairs of those popular functions is carefully established. Besides illustrating the effectiveness of the developed methods with two illustrative examples, the applicability are further discussed in the fractional order case.