Description
Book SynopsisThis book arose out of original research on the extension of well-established applications of complex numbers related to Euclidean geometry and to the space-time symmetry of two-dimensional Special Relativity. The system of hyperbolic numbers is extensively studied, and a plain exposition of space-time geometry and trigonometry is given. Commutative hypercomplex systems with four unities are studied and attention is drawn to their interesting properties.
Trade ReviewFrom the reviews: “It is worth pointing out that the book is mainly a text about commutative hypercomplex numbers and some of their applications to a 2-dimensional Minkowski spacetime. … This book should be interesting to anybody who is interested in applications of hypercomplex numbers … . In conclusion, I recommend this book to anyone who wants to learn about hypercomplex numbers.” (Emanuel Gallo, Mathematical Reviews, Issue 2010 d)
Table of ContentsThe Mathematics of Minkowski Space-Time: 1 N-Dimensional Hypercomplex Numbers and the associated Geometries.- Commutative Hypercomplex Number Systems.- The General Two-Dimensional System.- Linear Transformations and Geometries.- The Geometries Associated with Hypercomplex Numbers.- Conclusions.- 2 Trigonometry in the Minkowski Plane.- Geometrical Representation of Hyperbolic Numbers.- Basics of Hyperbolic Trigonometry.- Geometry in Pseudo-Euclidean Cartesian Plane.- Trigonometry in the Pseudo-Euclidean Plane.- Theorems on Equilateral Hyperbolas in the Pseudo-Euclidean Plane.- Some Examples of Triangle Solutions in the Minkowski Plane.- Conclusions.- 3 Uniform and Accelerated Motions in the Minkowski Space-Time (Twin Paradox).- Inertial Motions.- Inertial and Uniformly Accelerated Motions.- Non-uniformly Accelerated Motions.- Conclusions.- 4 General Two-Dimensional Hypercomplex Numbers.-Geometrical Representation.- Geometry and Trigonometry in Two-Dimensional Algebras.- Some Properties of Fundamental Conic Section.- Numerical Examples.- 5 Functions of a Hyperbolic Variable.- Some Remarks on Functions of a Complex Variable.- Functions of Hypercomplex Variables.- The Functions of a Hyperbolic Variable.- The Elementary Functions of a Canonical Hyperbolic Variable.- H-Conformal Mappings.- Commutative Hypercomplex Systems with Three Unities.- 6 Hyperbolic Variables on Lorentz Surfaces.- Introduction.- Gauss: Conformal Mapping of Surfaces.- Extension of Gauss Theorem: Conformal Mapping of Lorentz Surfaces.- Beltrami: Complex Variables on a Surface.- Beltrami’s Integration of Geodesic Equations.- Extension of Beltrami’s Equation to Non-Definite Differential Forms.- 7 Constant Curvature Lorentz Surfaces.- Introduction.- Constant Curvature RiemannSurfaces.- Constant Curvature Lorentz Surfaces.- Geodesics and Geodesic Distances on Riemann and Lorentz Surfaces.- Conclusions.- 8 Generalization of Two-Dimensional Special Relativity (Hyperbolic Transformations and the Equivalence Principle).- Physical Meaning of Transformations by Hyperbolic Functions.- Physical Interpretation of Geodesics on Riemann and Lorentz Surfaces with Positive Constant Curvature.- Einstein’s Way to General Relativity.- Conclusions.- II An Introduction to Commutative Hypercomplex Numbers.- 9 Commutative Segre’s Quaternions.- Introduction.- Hypercomplex Systems with Four Units.- Historical Introduction of Segre’s Quaternion.- Algebraic Properties of Commutative Quaternions.- Functions of a Quaternion Variable.- Mapping by Means of Quaternion Functions.- Elementary Functions of the Quaternions.- Elliptic-Hyperbolic Quaternions.- Elliptic-Parabolic Generalized Segre’s Quaternions.- 10 Constant Curvature Segre’s Quaternion Spaces.- Introduction.- Quaternion differential geometry and geodesic equations.- Orthogonality in Segre’s Quaternion Space.- Constant Curvature Quaternion Spaces.- Geodesic Equations in Quaternion Space.- Beltrami’s Integration Method for Quaternion Spaces.- Beltrami’s Integration Method for Quaternion Spaces.- Conclusions.- 11 A Matrix Formalization for Commutative Hypercomplex Systems.- Mathematical Operations.- Properties of the Characteristic Matrix M.- Functions of Hypercomplex Variable.- Functions of a Two-Dimensional Hypercomplex Variable.- Derivatives of a Hypercomplex Function.- Characteristic Differential Equation.- A Equivalence Between the Formalizations of Hypercomplex Numbers.
