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" Generalized trigonometric and hyperbolic functions / "
Ronald E. Mickens.
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BL
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841814
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Mickens, Ronald E.,1943-
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Generalized trigonometric and hyperbolic functions /\ Ronald E. Mickens.
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| Publication Statement
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Boca Raton, Florida :: CRC Press,, [2019]
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, ©2019
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1 online resource
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| ISBN
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0429446233
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: 0429821077
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: 0429821085
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: 0429821093
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: 9780429446238
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: 9780429821073
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: 9780429821080
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: 9780429821097
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1138333018
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9781138333017
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Includes bibliographical references.
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| Contents
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Cover; Half Title; Title Page; Copyright Page; Table of Contents; Dedication; List of Figures; Preface; Author; 1: TRIGONOMETRIC AND HYPERBOLIC SINE AND COSINE FUNCTIONS; 1.1 INTRODUCTION; 1.2 SINE AND COSINE: GEOMETRIC DEFINITIONS; 1.3 SINE AND COSINE: ANALYTIC DEFINITION; 1.3.1 Derivatives; 1.3.2 Integrals; 1.3.3 Taylor Series; 1.3.4 Addition and Subtraction Rules; 1.3.5 Product Rules; 1.4 SINE AND COSINE: DYNAMIC SYSTEM APPROACH; 1.4.1 x-y Phase-Space; 1.4.2 Symmetry Properties of Trajectories in Phase-Space; 1.4.3 Null-Clines; 1.4.4 Geometric Proof that All Trajectories Are Closed
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1.5 HYPERBOLIC SINE AND COSINE: DERIVED FROM SINE AND COSINE1.6 HYPERBOLIC FUNCTIONS: DYNAMIC SYSTEM DERIVATION; 1.7 [Theta]-PERIODIC HYPERBOLIC FUNCTIONS; 1.8 DISCUSSION; Notes and References; 2: ELLIPTIC FUNCTIONS; 2.1 INTRODUCTION; 2.2 [Theta]-PERIODIC ELLIPTIC FUNCTIONS; 2.3 ELLIPTIC HAMILTONIAN DYNAMICS; 2.4 JACOBI, CN, SN, AND DN FUNCTIONS; 2.4.1 Elementary Properties of Jacobi Elliptic Functions; 2.4.2 First Derivatives; 2.4.3 Differential Equations; 2.4.4 Calculation of u(theta) and the Period for cn, sn, dn; 2.4.5 Special Values of Jacobi Elliptic Functions
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2.5 ADDITIONAL PROPERTIES OF JACOBI ELLIPTIC FUNCTIONS2.5.1 Fundamental Relations for Square of Functions; 2.5.2 Addition Theorems; 2.5.3 Product Relations; 2.5.4 cn, sn, dn for Special k Values; 2.5.5 Fourier Series; 2.6 DYNAMICAL SYSTEM INTERPRETATION OF ELLIPTIC JACOBI FUNCTIONS; 2.6.1 Definition of the Dynamic System; 2.6.2 Limits k → 0+ and k → 1-; 2.6.3 First Integrals; 2.6.4 Bounds and Symmetries; 2.6.5 Second-Order Differential Equations; 2.6.6 Discussion; 2.7 HYPERBOLIC ELLIPTIC FUNCTIONS AS A DYNAMIC SYSTEM; 2.8 HYPERBOLIC Theta-PERIODIC ELLIPTIC FUNCTIONS; 2.9 DISCUSSION
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| Abstract
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Generalized Trigonometric and Hyperbolic Functions highlights, to those in the area of generalized trigonometric functions, an alternative path to the creation and analysis of these classes of functions. Previous efforts have started with integral representations for the inverse generalized sine functions, followed by the construction of the associated cosine functions, and from this, various properties of the generalized trigonometric functions are derived. However, the results contained in this book are based on the application of both geometrical phase space and dynamical systems methodologies. Features Clear, direct construction of a new set of generalized trigonometric and hyperbolic functions Presentation of why x2+y2 = 1, and related expressions, may be interpreted in three distinct ways All the constructions, proofs, and derivations can be readily followed and understood by students, researchers, and professionals in the natural and mathematical sciences
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Exponential functions.
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Hyperbola.
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Trigonometry.
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Exponential functions.
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Hyperbola.
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MATHEMATICS-- Arithmetic.
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MATHEMATICS-- Functional Analysis.
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MATHEMATICS-- Geometry-- General.
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MATHEMATICS-- Number Theory.
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Trigonometry.
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| Dewey Classification
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516.24/6
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| LC Classification
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QA531.M5845 2019
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