On Some New Properties of Quaternion Functions

Abstract

Quaternions discovered by W. R. Hamilton made a great contribution to the progress in noncommutative algebra and vector analysis. However, the analysis of quaternion functions has not been duly developed. The matter is that the notion of a derivative of quaternion functions of a quaternion variable has not been known until recently. The author has succeeded in improving the situation. The present work contains an account of the results obtained by him in this direction. The notion of an ℍ-derivative is introduced for quaternion functions of a quaternion variable. The existence of an ℍ-derivative of elementary functions is established retaining the well-known formulas for the corresponding functions from complex (real) analysis. The rules on the ℍ-differentiation of a sum, a product, and an inverse function are formulated and proved. Necessary and sufficient conditions for the existence of an ℍ-derivative are established. The notions of ℂ²-differentiation and ℂ²-holomorphy are introduced for quaternion functions of a quaternion variable. Three equivalent conditions are found, each of them being a necessary and sufficient one for ℂ²-differentiation. Representations by an integral and a power series are given for ℂ²-holomorphic functions. It is proved that the harmonicity of functions f(z), z · f(z), and f(z) · z is the necessary and sufficient condition for a function f to be Fueter-regular.