The exponential function of a quaternion variable

Abstract

In hypercomplex function theory one has always looked for constructing elementary functions, especially an exponential function. In this connection we can refer e.g. to the exponential function with values in the commutative algebra of quaternary numbers of M. Futagawa [8], in the commutative algebra of A. Douglis [5], as it appears in the works of R. P. Gilbert and G. Hile [9] and H. Begehr and R. P. Gilbert [1], and in the commutative infinite dimensional algebra of P. W. Ketchum [11] as it is constructed by H. H. Snyder 1121.

For the non commutative case we mention e.g, the exponential function of hyperanalytic reduced quaternions of R. Fueter [6] and that of E. Gollnitz [lo] in analytic quaternion function theory in Weierstrass' sense. The aim of this paper is to construct an exponential function within the framework of the hypercomplex function theory of quaternion variables as developed by the author in [2], [3] and [4]. This exponential function exp is two-sided- (1)-entire and satisfies the fundamental property for all x and y in . Moreover a quaternion-derivative d/dx is defined such that .

Additional information

Notes on contributors

F. F. Brackx