Cauchy-like Integral Theorems for Quaternion and Biquaternion Functions

Abstract

It is well known that there is an integral theorem for quaternion-valued functions analogous to Cauchys Theorem for complex-valued functions, namely Fueters Theorem. The class of quaternionic functions for which this applies are generally referred to as regular functions, and these provide the most productive means of generalising the class of holomorphic complex functions. This paper derives a second integral theorem, also analogous to Cauchys Theorem, and which is believed to be quite distinct from that of Fueter, despite appearances. The paper takes the opportunity to present the basis of the derivation of both theorems, and also their extension to the associated classes of right-regular and conjugate regular functions. Both theorems can also be extended into the biquaternionic domain in which the four quaternion coordinates may be complex valued. This is of interest in physics as the Hermitian biquaternions have a natural norm which is Minkowskian and provide an elegant formalism for Lorentz transformations.