Quaternionic slice regular functions and quaternionic Laplace transforms

TL;DR

This paper explores quaternionic slice regular functions and extends the classical Laplace transform to quaternions, revealing their relationships and properties in the quaternionic setting.

Contribution

It introduces a natural generalization of the Laplace transform to quaternions and clarifies the relations among different types of slice regular functions.

Findings

Left and right slice regular functions are related by an involution.

Quaternionic Laplace transforms preserve classical properties.

Two distinct quaternionic Laplace transforms are defined and analyzed.

Abstract

The functions studied in the paper are quaternion-valued functions of a quaternionic variable. It is show that the left slice regular functions and right slice regular functions are related by a particular involution. The relation between left slice regular functions, right slice regular functions and intrinsic regular functions is revealed. The classical Laplace transform can be naturally generalized to quaternions in two different ways, which transform a quaternion-valued function of a real variable to a left or right slice regular quaternion-valued function of a quaternionic variable. The usual properties of the classical Laplace transforms are generalized to quaternionic Laplace transforms.