Differentiable functions of Cayley-Dickson numbers and line integration

By S. Ludkovsky. 2004

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We investigate the superdifferentiability of functions defined on domains of the real octonion (Cayley) algebra and obtain a noncommutative version of the Cauchy-Riemann conditions. Then we study the noncommutative analogue of the Cauchy integral and criteria for functions of an octonion variable to be analytic. In particular, the octonion-valued exponential and logarithmic functions are considered. Moreover, superdifferentiable functions of variables belonging to Cayley-Dickson algebras (containing the octonion algebra as a proper subalgebra) of finite and infinite dimensions are investigated. Among the main results, the Cayley-Dickson-algebra analogues of the Cauchy, Hurewicz, Mittag-Löffler, Rouche, and Weierstrass theorems and the argument principle are proved. Examples of special functions such as the beta and gamma functions of Cayley-Dickson numbers are studied. Applications to the study of zeros of polynomials of Cayley-Dickson numbers are given.

Subfields: Applied Mathematics, Mathematical Physics

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