Complex Numbers in n Dimensions

By S. Olariu

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 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.

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.

• Language: English
• Publisher: Elsevier Science
• Release date: Jun 20, 2002
• ISBN: 9780080529585

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Agata

Preface

A regular, two-dimensional complex number x + iy can be represented geometrically by the modulus ρ = (x² + y²)¹/² and by the polar angle θ = arctan(y/x). The modulus ρ is multiplicative and the polar angle θ is additive upon the multiplication of ordinary complex numbers.

The quaternions of Hamilton are a system of hypercomplex numbers defined in four dimensions, the multiplication being a noncommutative operation, [1] and many other hypercomplex systems are possible, [2]–[4] but these interesting hypercomplex systems do not have all the required properties of regular, two-dimensional complex numbers which rendered possible the development of the theory of functions of a complex variable.

Two distinct systems of hypercomplex numbers in n dimensions will be described in this work, 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. [5] The n-complex numbers described in this work have the form u = x0+h1x1+…+hn−1xn−1 where h1,…hn−1 are the hypercomplex bases and the variables x0,…,xn−1 are real numbers, unless otherwise stated. If the n-complex number u is represented by the point A of coordinates x0,x1,…,xnand of n − 1 angular variables.

The first type of hypercomplex numbers described in this work is characterized by the presence, in an even number of dimensions n ≥ 4, of two polar axes, and by the presence, in an odd number of dimensions, of one polar axis. Therefore, these numbers will be called polar hypercomplex numbers in n dimensions. One polar axis is the normal through the origin O to the hyperplane v+ = 0, where v+ = x0 + x1 + …+ xn−1. In an even number n of dimensions, the second polar axis is the normal through the origin O to the hyperplane v− = 0, where v− = 0, where v+ = x0 + x1 + …+xn−1. Thus, in addition to the distance d, the position of the point A can be specified, in an even number of dimensions, by 2 polar angles θ+, θ−, by n/2 − 2 planar angles ψk, and by n/2 − 1 aziniuthal angles ϕk. In an odd nnmber of dimensions, the position of the point A is specified by d, by 1 polar angle θ+, by (n − 3)/2 planar angles ψk−1, and by (n − 1)/2 azinuithal angles ϕk. The multiplication rules for the polar hypercomplex bases h1,…hn−1 are hjhk = hj + k if 0 ≤ j + k ≤ n − 1, and hjhk = hj + k−n if n ≤ j + k ≤ 2n − 2, where h0 = 1.

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. The position of the point A is specified, in addition to the distance d, by n/2 − 1 planar angles ψk and by n/2 aziniuthal angles ϕk. These numbers will be called planar hypercomplex numbers. The multiplication rules for the planar hypercomplex bases h1,…,hn − 1 are hjhk = hj+k if 0 ≤ j + k ≤ n − 1, and hjhk = −hj+k−n if n ≤ j + k ≤ 2n − 2, where h0 = 1. 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. The azinuithal angles ϕk, 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 work n-dimensional cosexponential functions of the polar and respectively planar type. The polar cosexponential functions are a generalization to n dimensions of the hyperbolic functions cosh y, sinh y, and tlie planar cosexponential functions are a generalization to n dimensions of the trigonometric functions cos y, sin y. Addition theorems and other relations are obtained for the n-dimensional cosexponential functions.

Many of the properties of 2-dimensional complex functions can be extended to hypercomplex numbers in n dimensions. Thus, the functions f(u) of an n-complex variable which are defined by power series have derivatives independent of the direction of approach to the point under consideration. If the n-complex function f(u) of the n-complex variable u is written in terms of the real functions Pk(x0,…,xn−1), k = 0,…,n − 1, then relations of equality exist between the partial derivatives of the functions Pkof an n-complex function between two points A, B is independent of the path connecting A, B, in regions where f is regular. If f(uis expressed in this work in terms of the n-dimensional hypercomplex residue f(u0).

In the case of polar complex numbers, a polynomial can be written as a product of linear or quadratic factors, although 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 work 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 work is the interplay between the algebraic, the geometric and the analytic facets of the relations.

Chapter 1

Hyperbolic Complex Numbers in Two Dimensions

A system of hypercomplex numbers in 2 dimensions is described in this chapter, for which the multiplication is associative and commutative, and for which an exponential form and the concepts of analytic twocomplex function and contour integration can be defined. The twocomplex numbers introduced in this chapter have the form u = x + δy, the variables x, y being real numbers. The multiplication rules for the complex units 1, δ are 1 · δ = δ, δ² = 1. In a geometric representation, the twocomplex number u is represented by the point A of coordinates (x, y). The product of two twocomplex numbers is equal to zero if both numbers are equal to zero, or if one of the twocomplex numbers lies on the line x = y and the other on the line x = −y.

The exponential form of a twocomplex number, defined for x+y > 0, x − y > 0, is u = ρ exp(δλ/2), where the amplitude is ρ = (x² − y²)¹/² and the argument is λ = ln tan θ, tan θ = (x + y)/(x − y), 0 < θ < π/2. The trigonometric form of a twocomplex number is

, where d² = x² + y². The amplitude ρ is equal to zero on the lines x = ±y. The division 1/(x+δy) is possible provided that ρ ≠ 0. If u1 = x1+δy1, u2 = x2 + δy2 are twocomplex numbers of amplitudes and arguments ρ1, λ1 and respectively ρ2, λ2, then the amplitude and the argument ρ, λ of the product twocomplex number u1u2 = x1x2 + y1y2 + δ(x1y2 + y1x2) are ρ = ρ1ρ2, λ = λ1 + λ2. Thus, the amplitude ρ is a multiplicative quantity and the argument λ is an additive quantity upon the multiplication of twocomplex numbers, which reminds the properties of ordinary, two-dimensional complex numbers.

Expressions are given for the elementary functions of twocomplex variable. Moreover, it is shown that the region of convergence of series of powers of twocomplex variables is a rectangle having the sides parallel to the bisectors x = ±y.

A function f(u) of the twocomplex variable u = x + δy can be defined by a corresponding power series. It will be shown that the function f has a derivative limu→u0[f (u) − f(u0)]/(u − u0) independent of the direction of approach of u to u0. If the twocomplex function f(u) of the twocomplex variable u is written in terms of the real functions P(x,y), Q(x,y) of real variables x, y as f(u) = P(x,y) + δQ(x,y), then relations of equality exist between partial derivatives of the functions P,Q, and the functions P,Q are solutions of the two-dimensional wave equation.

of a twocomplex function between two points A,B is independent of the path connecting the points A,B.

A polynomial un + a1un−1 + an−1u + an can be written as a product of linear or quadratic factors, although the factorization may not be unique.

The twocomplex numbers described in this chapter are a particular case for n = 2 of the polar complex numbers in n dimensions discussed in Sec. 6.1.

1.1 Operations with hyperbolic twocomplex numbers

A hyperbolic complex number in two dimensions is determined by its two components (x, y). The sum of the hyperbolic twocomplex numbers (x, y) and (x′,y′) is the hyperbolic twocomplex number (x + x′,y + y′). The product of the hyperbolic twocomplex numbers (x,y) and (x′,y′) is defined in this chapter to be the hyperbolic twocomplex number (xx′ + yy′, xy′ + yx′).

Twocomplex numbers and their operations can be represented by writing the twocomplex number (x, y) as u = x + δy, where δ is a basis for which the multiplication rules are

(1.1)

Two twocomplex numbers u = x + δy, u′ = x′ + δy′ are equal, u = u′, if and only if x = x′, y = y′. If u = x + δy,u′ = x′ + δy′ are twocomplex numbers, the sum u + u′ and the product uu′ defined above can be obtained by applying the usual algebraic rules to the sum (x + δy) + (x′ + δy′) and to the product (x + δy) (x′ + δy′), and grouping of the resulting