# Complex Numbers in n Dimensions

**By**

- S. Olariu, National Institute of Physics and Nuclear Engineering, Bucharest, Romania

Two distinct systems of hypercomplex numbers in n dimensions are introduced in this book, for which the multiplication is associative and commutative, and which are rich enough in properties such that exponential and trigonometric forms exist and the concepts of analytic n-complex function, contour integration and residue can be defined.

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 functionsof 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.

The book presents a detailed analysis of the hypercomplex numbers in 2, 3 and 4 dimensions, then presents the properties of hypercomplex numbers in 5 and 6 dimensions, and it continues with a detailed analysis of polar and planar hypercomplex numbers in n dimensions. The essence of this book is the interplay between the algebraic, the geometric and the analytic facets of the relations.

### Audience

University libraries. Libraries of research institutes for mathematics and physics. Departments of mathematics and physics of universities. Departments of mathematics and theoretical physics of research institutes.

### Book information

- Published: June 2002
- Imprint: NORTH-HOLLAND
- ISBN: 978-0-444-51123-2

### Reviews

The author presents descriptions of several hypercomplex systems for which the multiplication is associative and commutative. In each case some properties of the respective functions are given.

Mathematical Reviews

### Table of Contents

1 Hyperbolic Complex Numbers in Two Dimensions

1.1 Operations with hyperbolic twocomplex numbers.

1.2 Geometric representation of hyperbolic twocomplex numbers.

1.3 Exponential and trigonometric forms of a twocomplex number.

1.4 Elementary functions of a twocomplex variable.

1.5 Twocomplex power series.

1.6 Analytic functions of twocomplex variables.

1.7 Integrals of twocomplex functions.

1.8 Factorization of twocomplex polynomials.

1.9 Representation of hyperbolic twocomplex numbers by irreducible matrices.

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 variable.

3.1.5 Power series of circular fourcomplex variables.

3.1.6 Analytic functions of circular fourcomplex variables.

3.1.7 Integrals of functions of circular fourcomplex variables.

3.1.8 Factorization of circular fourcomplex polynomials.

3.1.9 Representation of circular fourcomplex numbers by irreducible matrices.

3.2 Hyperbolic complex numbers in four dimensions.

3.2.1 Operations with hyperbolic fourcomplex numbers.

3.2.2 Geometric representation of hyperbolic fourcomplex numbers.

3.2.3 Exponential form of a hyperbolic fourcomplex number.

3.2.4 Elementary functions of a hyperbolic fourcomplex variable.

3.2.5 Power series of hyperbolic fourcomplex variables.

3.2.6 Analytic functions of hyperbolic fourcomplex variables.

3.2.7 Integrals of functions of hyperbolic fourcomplex variables.

3.2.8 Factorization of hyperbolic fourcomplex polynomials.

3.2.9 Representation of hyperbolic fourcomplex numbers by irreducible matrices.

3.3 Planar complex numbers in four dimensions.

3.3.1 Operations with planar fourcomplex numbers.

3.3.2 Geometric representation of planar fourcomplex numbers.

3.3.3 The planar fourdimensional cosexponential functions.

3.3.4 The exponential and trigonometric forms of planar fourcomplex numbers.

3.3.5 Elementary functions of planar fourcomplex variables.

3.3.6 Power series of planar fourcomplex variables.

3.3.7 Analytic functions of planar fourcomplex variables.

3.3.8 Integrals of functions of planar fourcomplex variables.

3.3.9 Factorization of planar fourcomplex polynomials.

3.3.10 Representation of planar fourcomplex numbers by irreducible matrices.

3.4 Polar complex numbers in four dimensions.

3.4.1 Operations with polar fourcomplex numbers.

3.4.2 Geometric representation of polar fourcomplex numbers.

3.4.3 The polar fourdimensional cosexponential functions.

3.4.4 The exponential and trigonometric forms of a polar fourcomplex number.

3.4.5 Elementary functions of polar fourcomplex variables.

3.4.6 Power series of polar fourcomplex variables.

3.4.7 Analytic functions of polar fourcomplex variables.

3.4.8 Integrals of functions of polar fourcomplex variables.

3.4.9 Factorization of polar fourcomplex polynomials.

3.4.10 Representation of polar fourcomplex numbers by irreducible matrices.

4 Complex Numbers in 5 Dimensions

4.1 Operations with polar complex numbers in 5 dimensions.

4.2 Geometric representation of polar complex numbers in 5 dimensions.

4.3 The polar 5-dimensional cosexponential functions.

4.4 Exponential and trigonometric forms of polar 5-complex numbers.

4.5 Elementary functions of a polar 5-complex variable.

4.6 Power series of 5-complex numbers.

4.7 Analytic functions of a polar 5-complex variable.

4.8 Integrals of polar 5-complex functions.

4.9 Factorization of polar 5-complex polynomials.

4.10 Representation of polar 5-complex numbers by irreducible matrices

5 Complex Numbers in 6 Dimensions

5.1 Polar complex numbers in 6 dimensions.

5.1.1 Operations with polar complex numbers in 6 dimensions.

5.1.2 Geometric representation of polar complex numbers in 6 dimensions.

5.1.3 The polar 6-dimensional cosexponential functions.

5.1.4 Exponential and trigonometric forms of polar 6-complex numbers.

5.1.5 Elementary functions of a polar 6-complex variable.

5.1.6 Power series of polar 6-complex numbers.

5.1.7 Analytic functions of a polar 6-complex variable.

5.1.8 Integrals of polar 6-complex functions.

5.1.9 Factorization of polar 6-complex polynomials.

5.1.10 Representation of polar 6-complex numbers by irreducible matrices.

5.2 Planar complex numbers in 6 dimensions.

5.2.1 Operations with planar complex numbers in 6 dimensions.

5.2.2 Geometric representation of planar complex numbers in 6 dimensions.

5.2.3 The planar 6-dimensional cosexponential functions.

5.2.4 Exponential and trigonometric forms of planar 6-complex numbers.

5.2.5 Elementary functions of a planar 6-complex variable.

5.2.6 Power series of planar 6-complex numbers.

5.2.7 Analytic functions of a planar 6-complex variable.

5.2.8 Integrals of planar 6-complex functions.

5.2.9 Factorization of planar 6-complex polynomials.

5.2.10 Representation of planar 6-complex numbers by irreducible matrices.

6 Commutative Complex Numbers in n Dimensions

6.1 Polar complex numbers in n dimensions.

6.1.1 Operations with polar n-complex numbers.

6.1.2 Geometric representation of polar n-complex numbers.

6.1.3 The polar n-dimensional cosexponential functions.

6.1.4 Exponential and trigonometric forms of polar n-complex numbers.

6.1.5 Elementary functions of a polar n-complex variable.

6.1.6 Power series of polar n-complex numbers.

6.1.7 Analytic functions of polar n-complex variables.

6.1.8 Integrals of polar n-complex functions.

6.1.9 Factorization of polar n-complex polynomials.

6.1.10 Representation of polar n-complex numbers by irreducible matrices.

6.2 Planar complex numbers in even n dimensions.

6.2.1 Operations with planar n-complex numbers.

6.2.2 Geometric representation of planar n-complex numbers.

6.2.3 The planar n-dimensional cosexponential functions.

6.2.4 Exponential and trigonometric forms of planar n-complex numbers.

6.2.5 Elementary functions of a planar n-complex variable.

6.2.6 Power series of planar n-complex numbers.

6.2.7 Analytic functions of planar n-complex variables.

6.2.8 Integrals of planar n-complex functions.

6.2.9 Factorization of planar n-complex polynomials.

6.2.10 Representation of planar n-complex numbers by irreducible matrices.

Bibliography.

Index