Application of Exponential Kernel to Laplace Transform

Sami M Al-Jaber

doi:10.4236/jamp.2019.75075

Abstract

In this paper, the exponential decreasing kernel is used in Laplace integral transform to transform a function from a certain domain to another domain. It is shown, in a rigorous way, that the Laplace transform of the delta function is exactly one half rather than one, as it is believed. In addition, when this kernel is used in integral transform of attractive and repulsive Coulomb potential, it yields a finite definite value at the point of singularity.

Highlights

Kernels determine an implicit map that transforms a function or data from the input space to a feature space, and determine its distribution in the latter space
In a rigorous way, that the Laplace transform of the delta function is exactly one half rather than one, as it is believed. When this kernel is used in integral transform of attractive and repulsive Coulomb potential, it yields a finite definite value at the point of singularity
Finding the Laplace transform of a function and its properties is normally discussed in standard mathematical physics books [7] [8]

Summary

Introduction

Kernels determine an implicit map that transforms a function or data from the input space to a feature space, and determine its distribution in the latter space. We consider a function with repulsive Coulomb-like form on the positive real axis, and with attractive Coulomb-like form on the negative real axis This function is singular at the origin, and its right-hand and left-hand limits towards the origin are +∞ and −∞ respectively. It is shown, with this decreasing exponential kernel, that the value of this function is exactly zero which is the average between its limiting values at the origin. The last section of this paper is devoted for conclusion and discussion

The Laplace Transform of Delta-Function

Application of the Exponential Kernel to Coulomb-Like Function

Conclusion and Discussion