Calculus with a quaternionic variable

Abstract

Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn in trying to extend our reach to include quaternions. The noncommutativity of the quaternion algebra poses obstacles for the usual manipulations of calculus, but we show in this paper how many of those obstacles can be overcome. The surprising result is that the first order term in the expansion of F(x+δ) is a compact formula involving both F′(x) and [F(x)−F(x∗)]/(x−x∗). This advance in the differential calculus for quaternionic variables also leads us to some progress in studying integration.