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Differentiable functions of Cayley-Dickson numbers

By S. V. Ludkovsky and F. Van Oystaeyen


We investigate superdifferentiability of functions defined on regions of the real octonion (Cayley) algebra and obtain a noncommutative version of the Cauchy-Riemann conditions. Then we study the noncommutative analog of the Cauchy integral as well as criteria for functions of an octonion variable to be analytic. In particular, the octonion exponential and logarithmic functions are being considered. Moreover, superdifferentiable functions of variables belonging to Cayley-Dickson algebras (containing the octonion algebra as the proper subalgebra) finite and infinite dimensional are investigated. Among main results there are the Cayley-Dickson algebras analogs of Caychy’s theorem, Hurtwitz’, argument principle, Mittag-Leffler’s, Rouche’s and Weierstrass’ theorems

Year: 2012
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