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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
1.5 HYPERBOLIC SINE AND COSINE: DERIVED FROM SINE AND COSINE
1.6 HYPERBOLIC FUNCTIONS: DYNAMIC SYSTEM DERIVATION
1.7 0-PERIODIC HYPERBOLIC FUNCTIONS
1.8 DISCUSSION
Notes and References
2 ELLIPTIC FUNCTIONS
2.1 INTRODUCTION
2.2 0-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(0) and the Period for cn, sn, dn
2.4.5 Special Values of Jacobi Elliptic Functions
2.5 ADDITIONAL PROPERTIES OF JACOBI ELLIPTIC FUNCTIONS
2.5.1 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 Limitsk→0 andk→l-
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 0-PERIODIC ELLIPTIC FUNCTIONS
2.9 DISCUSSION
Notes and References
3 SQUARE FUNCTIONS
3.1 INTRODUCTION
3.2 PROPERTIES OF THE SQUARE TRIGONOMETRIC FUNCTIONS
3.3 PERIOD OF THE SQUARE TRIGONOMETRIC FUNCTIONS IN THE VARIABLE
3.4 FOURIER SERIES OF THE SQUARE TRIGONOMETRIC FUNCTIONS
3.5 DYNAMIC SYSTEM INTERPRETATION OF£üx£ü £üy£ü = 1
3.6 HYPERBOLIC SQUARE FUNCTIONS: DYNAMICS SYSTEM APPROACH
3.7 PERIODIC HYPERBOLIC SQUARE FUNCTIONS
Notes and References
4 PARABOLIC TRIGONOMETRIC FUNCTIONS
4.1 INTRODUCTION
4.2 H(x, y) =£üy£ü (1/2) x2 AS A DYNAMIC SYSTEM
4.3 GEOMETRIC ANALYSIS OF£üy£ü (1/2)x2 = 1/2=1 AS A DYNAMIC SYSTEM
4.4 lyl-(1/2)x2=1/2
4.5 GEOMETRIC ANALYSIS OF£üy£ü (1/2)x2 =1/2
Notes and References
5 GENERALIZED PERIODIC SOLUTIONS OF f(t)2 g(t)2 = 1
5.1 INTRODUCTION
5.2 GENERALIZED COSINE AND SINE FUNCTIONS
5.3 MATHEMATICAL STRUCTURE OF O(t)
5.4 AN EXAMPLE: A(t)=alsin(2π/T)t
5.5 DIFFERENTIAL EQUATION FOR f(t) AND g(t)
5.6 DISCUSSION
5.7 NON-PERIODIC SOLUTION; OF f2(t) g2(t)=1
Notes and References
6 RESUME OF (SOME) PREVIOUS RESULTS ON GENERALIZED TRIGONOMETRIC FUNCTIONS
6.1 INTRODUCTION
6.2 DIFFERENTIAL EQUATION FORMULATION
6.3 DEFINITION AS INTEGRAL FORMS
6.4 GEOMETRIC APPROACH
6.5 SYMMETRY CONSIDERATIONS AND CONSEQUENCES
6.5.1 Symmetry Transformation and Consequences
6.5.2 Hamiltonian Formulation
6.5.3 Area of Enclosed Curve
6.5.4 Period
6.6 SUMMARY
Notes and References
7 GENERALIZED TRIGONOMETRIC FUNCTIONS:£üy£üp £üx£üq=1
7.1 INTRODUCTION
7.2 METHODOLOGY
7.3 SUMMARY
7.4 GALLERY OF PARTICULAR SOLUTIONS
8 GENERALIZED TRIGONOMETRIC HYPERBOLIC FUNCTIONS:£üy£üp £üx£üq=1
8.1 INTRODUCTION
8.2 SOLUTIONS
8.3 GALLERY OF SPECIAL SOLUTIONS
9 APPLICATIONS AND ADVANCED TOPICS
9.1 INTRODUCTION
9.2 ODD-PARITY SYSTEMS AND THEIR FOURIER REPRESENTATIONS
9.3 TRULY NONLINEAR OSCILLATORS
9.3.1 Antisymmetric, Constant Force Oscillator
9.3.2 Particle in a Box
9.3.3 Restricted Duffing Equation
9.4 ATEB PERIODIC FUNCTIONS
9.5 EXACT DISCRETIZATION OF THE JACOBI ELLIPTI