A Commutative Hypercomplex Algebra with Associated Function Theory

Abstract

The group ring ℂ² of pairs (ξ,η) of classical complex numbers is used as an algebraic basis for a function theory for one four-dimensional variable of the form Z = 1 x + i y + j z + k ct. All of the axioms and properties of the complex analysis are carried forward into four dimensions, including functions, derivatives, integrals, and even the notation. The new analysis properly subsumes and extends the classical complex analysis. The single unexpected divergence from traditional properties is that the sets (ξ, 0) and (0, η) of non invertible elements are interpreted as the zero element (zero set). The zero set can be constrained to lie in the relativistic limit of special relativity; consequently, the new analysis scheme may be applied in any real-world, physical problem. The four-dimensional Cauchy-Riemann equations provide a way to apply continuity conditions to the field equations of physics, while at the same time converting PDEs to easier-to-solve ODEs. Significant simplifications are achieved for many physics field equations. The technique is illustrated on the Emden’s equation. Symbolic manipulations in four dimensions are easily handled by any computer algebra program that can perform classical complex algebraic operations.