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**Complex numbers in \(N\) dimensions.**
*(English)*
Zbl 1023.30001

North-Holland Mathematics Studies. 190. Amsterdam: North-Holland. xvi, 269 p. (2002).

This work is devoted to the properties of two distinct systems of hypercomplex numbers. For these systems the associative and commutative multiplication exists. Moreover, for such numbers exponential and trigonometric forms are introduced. It is important, that in corresponding versions of hypercomplex analysis the concepts of analytic function, contour integration and residue can be defined.

The first type of hypercomplex numbers described in this work is characterized by the presence, in an even number of dimensions \(n\geq 4\), of two polar axes, and by the pres-ence, in an odd number of dimensions, of one polar axis. These numbers are called polar hypercomplex numbers in n dimensions.

The other type of hypercomplex numbers described in this work exists as a distinct entity only when the number of dimensions \(n\) of the space is even. These numbers will be called planar hypercomplex numbers. For \(n=2\) the planar hypercomplex numbers become the usual \(2\)-dimensional complex numbers \(x+iy\).

The development of analytic functions of hypercomplex variables was rendered possible by the existence of an exponential form of the \(n\)-complex numbers. This leads to the notion of \(n\)-dimensional hypercomplex residue. The analogs for the elementary functions of \(n\)-complex variable are constructed. In particular, the generalizations to \(n\) dimensions of the hyperbolic functions \(\cosh y, \sinh y\) and of the trigonometric functions \( \cos y, \sin y\) are obtained.

Some important facts of \(\mathbb C^1\)-analysis extend to the hypercomplex situation described in the work under review. The author defines functions by power series, and such functions have derivatives independent of the direction of approach to the point. So, the problem of the independence of line integrals of the integration path is investigated. Besides, for an analytic \(n\)-complex function \(f\) the integral \(\oint_{\Gamma} f(u) du / (u-u_0) \) is expressed in terms of the \(n\)-dimensional hypercomplex residue \(f(u_0)\). This work contains sufficiently deep algebraic and geometric analysis of the properties of the hypercomplex system under consideration.

The first type of hypercomplex numbers described in this work is characterized by the presence, in an even number of dimensions \(n\geq 4\), of two polar axes, and by the pres-ence, in an odd number of dimensions, of one polar axis. These numbers are called polar hypercomplex numbers in n dimensions.

The other type of hypercomplex numbers described in this work exists as a distinct entity only when the number of dimensions \(n\) of the space is even. These numbers will be called planar hypercomplex numbers. For \(n=2\) the planar hypercomplex numbers become the usual \(2\)-dimensional complex numbers \(x+iy\).

The development of analytic functions of hypercomplex variables was rendered possible by the existence of an exponential form of the \(n\)-complex numbers. This leads to the notion of \(n\)-dimensional hypercomplex residue. The analogs for the elementary functions of \(n\)-complex variable are constructed. In particular, the generalizations to \(n\) dimensions of the hyperbolic functions \(\cosh y, \sinh y\) and of the trigonometric functions \( \cos y, \sin y\) are obtained.

Some important facts of \(\mathbb C^1\)-analysis extend to the hypercomplex situation described in the work under review. The author defines functions by power series, and such functions have derivatives independent of the direction of approach to the point. So, the problem of the independence of line integrals of the integration path is investigated. Besides, for an analytic \(n\)-complex function \(f\) the integral \(\oint_{\Gamma} f(u) du / (u-u_0) \) is expressed in terms of the \(n\)-dimensional hypercomplex residue \(f(u_0)\). This work contains sufficiently deep algebraic and geometric analysis of the properties of the hypercomplex system under consideration.

Reviewer: I.M.Mitelman (Odessa)