Quaternionic Slice Regular Functions and Quaternionic Laplace Transforms

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