Analysis Over Cayley-Dickson Numbers and Its Applications

The book is devoted to non-commutative analysis over Cayley-Dickson algebras and its applications to partial differential equations. In the first chapter super-differentiability of functions is described on regions of the real Cayley-Dickson algebra. The non-commutative analog of the Cauchy integral as well as criteria for functions of a Cayley-Dickson variable to be analytic are exposed. Among the main results we have the Cayley- Dickson algebras analogs of Caychy's theorem, Hurtwitz', argument principle, Mittag-Leffler's, Rouche's and Weierstrass' theorems, etc. In the second chapter the theory of meromorphic functions of the Cayley-Dickson variables is presented. Their properties and methods of calculations of their residues and arguments are described. The third chapter exposes material about differential equations over the Cayley-Dickson algebras. In chapter IV the technique for integration of partial differential equations with variable piecewise continuous or generalized coefficients is written. The fifth chapter contains results on the non- commutative multidimensional Laplace transforms.