Complex Numbers in n Dimensions (eBook)

S. Olariu (Autor)

eBook Download: PDF

2002 | 1. Auflage
286 Seiten
Elsevier Science (Verlag)
978-0-08-052958-5 (ISBN)

Infos

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.

Inhalt

Cover 1
Contents 11
Chapter 1. Hyperbolic Complex Numbers in Two Dimensions 17
1.1 Operations with hyperbolic twocomplex numbers 18
1.2 Geometric representation of hyperbolictwocomplex numbers 20
1.3 Exponential and trigonometric forms of a twocomplex number 22
1.4 Elementary functions of a twocomplex variable 24
1.5 Twocomplex power series 26
1.6 Analytic functions of twocomplex variables 28
1.7 Integrals of twocomplex functions 30
1.8 Factorization of twocomplex polynomials 31
1.9 Representation of hyperbolic twocomplex numbers by irreducible matrices 32
Chapter 2. Complex Numbers in Three Dimensions 33
2.1 Operations with tricomplex numbers 35
2.2 Geometric representation of tricomplex numbers 36
2.3 The tricomplex cosexponential functions 43
2.4 Exponential and trigonometric forms of tricomplex numbers 47
2.5 Elementary functions of a tricomplex variable 51
2.6 Tricomplex power series 54
2.7 Analytic functions of tricomplex variables 58
2.8 Integrals of tricomplex functions 60
2.9 Factorization of tricomplex polynomials 63
2.10 Representation of tricomplex numbers by irreducible matrices 66
Chapter 3. Commutative Complex Numbers in Four Dimensions 67
3.1 Circular complex numbers in four dimensions 70
3.2 Hyperbolic complex numbers in four dimensions 93
3.3 Planar complex numbers in four dimensions 109
3.4 Polar complex numbers in four dimensions 137
Chapter 4. Complex Numbers in 5 Dimensions 165
4.1 Operations with polar complex numbers in 5 dimensions 166
4.2 Geometric representation of polar complex numbers in 5 dimensions 167
4.3 The polar 5-dimensional cosexponential functions 170
4.4 Exponential and trigonometric forms of polar 5-complex numbers 175
4.5 Elementary functions of a polar 5-complex variable 177
4.6 Power series of 5-complex numbers 177
4.7 Analytic functions of a polar 5-complex variable 179
4.8 Integrals of polar 5-complex functions 179
4.9 Factorization of polar 5-complex polynomials 180
4.10 Representation of polar 5-complex numbers by irreducible matrices 181
Chapter 5. Complex Numbers in 6 Dimensions 183
5.1 Polar complex numbers in 6 dimensions 184
5.2 Planar complex numbers in 6 dimensions 197
Chapter 6. Commutative Complex Numbers in n Dimensions 211
6.1 Polar complex numbers in n dimensions 214
6.2 Planar complex numbers in even n dimensions 248
Bibliography 277
Index 279

Produkt-Details

• Erscheint lt. Verlag: 20.6.2002
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-10: 0-08-052958-5 / 0080529585
• ISBN-13: 978-0-08-052958-5 / 9780080529585