Complex Numbers In N Dimensions
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Complex Numbers In N Dimensions
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Author : S. Olariu
language : en
Publisher: Elsevier
Release Date : 2002-06-20
Complex Numbers In N Dimensions written by S. Olariu and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002-06-20 with Mathematics categories.
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.
Holomorphic Functions In The Plane And N Dimensional Space
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Author : Klaus Gürlebeck
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-11-16
Holomorphic Functions In The Plane And N Dimensional Space written by Klaus Gürlebeck and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-11-16 with Mathematics categories.
Complex analysis nowadays has higher-dimensional analoga: the algebra of complex numbers is replaced then by the non-commutative algebra of real quaternions or by Clifford algebras. During the last 30 years the so-called quaternionic and Clifford or hypercomplex analysis successfully developed to a powerful theory with many applications in analysis, engineering and mathematical physics. This textbook introduces both to classical and higher-dimensional results based on a uniform notion of holomorphy. Historical remarks, lots of examples, figures and exercises accompany each chapter.
Complex Numbers In N Dimensions
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Author : S. Olariu
language : en
Publisher: North Holland
Release Date : 2002-07-04
Complex Numbers In N Dimensions written by S. Olariu and has been published by North Holland this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002-07-04 with Mathematics categories.
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.
Complex Numbers The Higher Dimensional Forms 2nd Edition
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Author : Dennis Morris
language : en
Publisher: Createspace Independent Pub
Release Date : 2015-02-28
Complex Numbers The Higher Dimensional Forms 2nd Edition written by Dennis Morris and has been published by Createspace Independent Pub this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-02-28 with Science categories.
This book is about the higher dimensional complex numbers. Although it was not known when the first edition was printed, the higher dimensional complex numbers are spinor spaces and the classical spaces, including our 4-dimensional space-time, arise from these spinor spaces by superimposition of isomorphic algebras. This book covers the mathematical foundations underlying the mathematics of spinors and the space-time of classical physics.To understand this book, the reader needs no more than an elementary understanding of matrices and an elementary understanding of finite group theory. Previous knowledge of complex numbers would be helpful.
Complex Numbers In Geometry
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Author : I. M. Yaglom
language : en
Publisher: Academic Press
Release Date : 2014-05-12
Complex Numbers In Geometry written by I. M. Yaglom and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-05-12 with Mathematics categories.
Complex Numbers in Geometry focuses on the principles, interrelations, and applications of geometry and algebra. The book first offers information on the types and geometrical interpretation of complex numbers. Topics include interpretation of ordinary complex numbers in the Lobachevskii plane; double numbers as oriented lines of the Lobachevskii plane; dual numbers as oriented lines of a plane; most general complex numbers; and double, hypercomplex, and dual numbers. The text then takes a look at circular transformations and circular geometry, including ordinary circular transformations, axial circular transformations of the Lobachevskii plane, circular transformations of the Lobachevskii plane, axial circular transformations, and ordinary circular transformations. The manuscript is intended for pupils in high schools and students in the mathematics departments of universities and teachers' colleges. The publication is also useful in the work of mathematical societies and teachers of mathematics in junior high and high schools.
Complex Numbers And Vectors
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Author : Les Evans
language : en
Publisher: Aust Council for Ed Research
Release Date : 2006
Complex Numbers And Vectors written by Les Evans and has been published by Aust Council for Ed Research this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Electronic books categories.
Complex Numbers and Vectors draws on the power of intrigue and uses appealing applications from navigation, global positioning systems, earthquakes, circus acts and stories from mathematical history to explain the mathematics of vectors and the discoveries of complex numbers. The text includes historical and background material, discussion of key concepts, skills and processes, commentary on teaching and learning approaches, comprehensive illustrative examples with related tables, graphs and diagrams throughout, references for each chapter (text and web-based), student activities and sample solution notes, and an extensive bibliography.
Complex Numbers From A To Z
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Author : Titu Andreescu
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-10-08
Complex Numbers From A To Z written by Titu Andreescu and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-10-08 with Mathematics categories.
* Learn how complex numbers may be used to solve algebraic equations, as well as their geometric interpretation * Theoretical aspects are augmented with rich exercises and problems at various levels of difficulty * A special feature is a selection of outstanding Olympiad problems solved by employing the methods presented * May serve as an engaging supplemental text for an introductory undergrad course on complex numbers or number theory
Normal Partitions And Hierarchical Fillings Of N Dimensional Spaces
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Author : Zhizhin, Gennadiy Vladimirovich
language : en
Publisher: IGI Global
Release Date : 2020-12-25
Normal Partitions And Hierarchical Fillings Of N Dimensional Spaces written by Zhizhin, Gennadiy Vladimirovich and has been published by IGI Global this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-12-25 with Mathematics categories.
In the study of the structure of substances in recent decades, phenomena in the higher dimension was discovered that was previously unknown. These include spontaneous zooming (scaling processes), discovery of crystals with the absence of translational symmetry in three-dimensional space, detection of the fractal nature of matter, hierarchical filling of space with polytopes of higher dimension, and the highest dimension of most molecules of chemical compounds. This forces research to expand the formulation of the question of constructing n-dimensional spaces, posed by David Hilbert in 1900, and to abandon the methods of considering the construction of spaces by geometric figures that do not take into account the accumulated discoveries in the physics of the structure of substances. There is a need for research that accounts for the new paradigm of the discrete world and provides a solution to Hilbert's 18th problem of constructing spaces of higher dimension using congruent figures. Normal Partitions and Hierarchical Fillings of N-Dimensional Spaces aims to consider the construction of spaces of various dimensions from two to any finite dimension n, taking into account the indicated conditions, including zooming in on shapes, properties of geometric figures of higher dimensions, which have no analogue in three-dimensional space. This book considers the conditions of existence of polytopes of higher dimension, clusters of chemical compounds as polytopes of the highest dimension, higher dimensions in the theory of heredity, the geometric structure of the product of polytopes, the products of polytopes on clusters and molecules, parallelohedron and stereohedron of Delaunay, parallelohedron of higher dimension and partition of n-dimensional spaces, hierarchical filling of n-dimensional spaces, joint normal partitions, and hierarchical fillings of n-dimensional spaces. In addition, it pays considerable attention to biological problems. This book is a valuable reference tool for practitioners, stakeholders, researchers, academicians, and students who are interested in learning more about the latest research on normal partitions and hierarchical fillings of n-dimensional spaces.
Projective Geometry Of N Dimensions
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Author : Otto Schreier
language : en
Publisher:
Release Date : 1961
Projective Geometry Of N Dimensions written by Otto Schreier and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1961 with Algebra categories.
Complex Numbers
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Author : Noordin Jella
language : de
Publisher: LAP Lambert Academic Publishing
Release Date : 2012
Complex Numbers written by Noordin Jella and has been published by LAP Lambert Academic Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with categories.
A complex number is a number which can be put in the form , where and are real numbers and is called the imaginary unit. In this expression, is called the real part and the imaginary part of the complex number. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number can be identified with the point . A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with only real numbers. This has practically analyzed how to solve different problems concerning complex numbers such as Abraham De Moivre s Formula; Formula for Cosine and Sine Individually; Leonhard Euler s Formula; Products of Complex Numbers and De Moivre's Theorem; and different equations for practice. This Book is special for higher learning institutions worldwide.