- Print ISBN 9780444511232
- Electronic ISBN 9780080529585
The first type of hypercomplex numbers, called polar hypercomplex numbers, is characterized by the presence in an even number of dimensions greater or equal to 4 of two polar axes, and by the presence in an odd number of dimensions of one polar axis. The other type of hypercomplex numbers exists as a distinct entity only when the number of dimensions n of the space is even, and since the position of a point is specified with the aid of n/2-1 planar angles, these numbers have been called planar hypercomplex numbers.
The development of the concept of analytic functions of hypercomplex variables was rendered possible by the existence of an exponential form of the n-complex numbers. Azimuthal angles, which are cyclic variables, appear in these forms at the exponent, and lead to the concept of n-dimensional hypercomplex residue. Expressions are given for the elementary functions of n-complex variable. In particular, the exponential function of an n-complex number is expanded in terms of functions called in this book n-dimensional cosexponential functions of the polar and respectively planar type, which are generalizations to n dimensions of the sine, cosine and exponential functions.
In the case of polar complex numbers, a polynomial can be written as a product of linear or quadratic factors, although it is interesting that several factorizations are in general possible. In the case of planar hypercomplex numbers, a polynomial can always be written as a product of linear factors, although, again, several factorizations are in general possible.
2 Complex Numbers in Three Dimensions
2.1 Operations with tricomplex numbers.
2.2 Geometric representation of tricomplex numbers.
2.3 The tricomplex cosexponential functions.
2.4 Exponential and trigonometric forms of tricomplex numbers.
2.5 Elementary functions of a tricomplex variable.
2.6 Tricomplex power series.
2.7 Analytic functions of tricomplex variables.
2.8 Integrals of tricomplex functions.
2.9 Factorization of tricomplex polynomials.
2.10 Representation of tricomplex numbers by irreducible matrices.
3 Commutative Complex Numbers in Four Dimensions.
3.1 Circular complex numbers in four dimensions.
3.1.1 Operations with circular fourcomplex numbers.
3.1.2 Geometric representation of circular fourcomplex numbers.
3.1.3 The exponential and trigonometric forms of circular fourcomplex numbers.
3.1.4 Elementary functions of a circular fourcomplex vari