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Department of Mathematics, University of Michigan, Ann Arbr, Michigan 48109, USA

ABSTRACT

This paper generalizes the Cauchy-Riemann equations for complex differentiable functions to functions over certain n-dimensional commutative algebras over C with identity. Since the corresponding linear first order partial differential equations depend, u priori, on the basis chosen, this paper shows that in fact the set of functions which are differentiable is independent of the choice of basis. In the process, this paper generalizes to higher dimensions the notions of real and imaginary parts and of conjugation and the relationship between conjugation and multiplication for algebras over C².