  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 investigated. 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 Cauchy's theorem,
Hurwitz', argument principle, Mittag-Leffler's, Rouche's and Weierstrass'
theorems.
