Atjaunināt sīkdatņu piekrišanu

E-grāmata: Generalized Trigonometric and Hyperbolic Functions [Taylor & Francis e-book]

(Clark Atlanta University, SW Atlanta, Georgia, USA)
  • Formāts: 194 pages, 47 Illustrations, black and white
  • Izdošanas datums: 14-Jan-2019
  • Izdevniecība: CRC Press
  • ISBN-13: 9780429446238
Citas grāmatas par šo tēmu:
  • Taylor & Francis e-book
  • Cena: 231,23 €*
  • * this price gives unlimited concurrent access for unlimited time
  • Standarta cena: 330,33 €
  • Ietaupiet 30%
Generalized Trigonometric and Hyperbolic FunctionsPalielināt
  • Formāts: 194 pages, 47 Illustrations, black and white
  • Izdošanas datums: 14-Jan-2019
  • Izdevniecība: CRC Press
  • ISBN-13: 9780429446238
Citas grāmatas par šo tēmu:
Taylor & Francis e-booksMore info about Taylor & Francis e-books
Rokasgrāmatas mājas lapa: https://www.taylorfrancis.com/books/9780429446238
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
Dedication v
List of Figures
xi
Preface xiii
Author xvii
1 Trigonometric and Hyperbolic Sine and Cosine Functions
1(22)
1.1 Introduction
1(1)
1.2 Sine and Cosine: Geometric Definitions
2(1)
1.3 Sine and Cosine: Analytic Definition
3(5)
1.3.1 Derivatives
5(1)
1.3.2 Integrals
6(1)
1.3.3 Taylor Series
6(1)
1.3.4 Addition and Subtraction Rules
7(1)
1.3.5 Product Rules
7(1)
1.4 Sine and Cosine: Dynamic System Approach
8(6)
1.4.1 x-y Phase-Space
9(1)
1.4.2 Symmetry Properties of Trajectories in Phase-Space
10(1)
1.4.3 Null-Clines
10(3)
1.4.4 Geometric Proof that All Trajectories Are Closed
13(1)
1.5 Hyperbolic Sine and Cosine: Derived From Sine and Cosine
14(3)
1.6 Hyperbolic Functions: Dynamic System Derivation
17(1)
1.7 θ-Periodic Hyperbolic Functions
18(2)
1.8 Discussion
20(3)
Notes and References
22(1)
2 Elliptic Functions
23(24)
2.1 Introduction
23(1)
2.2 Aperiodic Elliptic Functions
23(3)
2.3 Elliptic Hamiltonian Dynamics
26(2)
2.4 Jacobi, CN, SN, and DN Functions
28(6)
2.4.1 Elementary Properties of Jacobi Elliptic Functions
29(1)
2.4.2 First Derivatives
30(1)
2.4.3 Differential Equations
31(1)
2.4.4 Calculation of u(θ) and the Period for cn, sn, tin
32(1)
2.4.5 Special Values of Jacobi Elliptic Functions
33(1)
2.5 ADDITIONAL PROPERTIES OF JACOBI ELLIPTIC FUNCTIONS
34(3)
2.5.1 Fundamental Relations for Square of Functions
34(1)
2.5.2 Addition Theorems
34(2)
2.5.3 Product Relations
36(1)
2.5.4 cn, sn, dn for Special k Values
36(1)
2.5.5 Fourier Series
36(1)
2.6 Dynamical System Interpretation of Elliptic Jacobi Functions
37(3)
2.6.1 Definition of the Dynamic System
37(1)
2.6.2 Limits k → 0+ and k → 1-
38(1)
2.6.3 First Integrals
38(1)
2.6.4 Bounds and Symmetries
39(1)
2.6.5 Second-Order Differential Equations
40(1)
2.6.6 Discussion
40(1)
2.7 Hyperbolic Elliptic Functions as a Dynamic System
40(1)
2.8 Hyperbolic θ-Periodic Elliptic Functions
41(4)
2.9 Discussion
45(2)
Notes and References
45(2)
3 Square Functions
47(20)
3.1 Introduction
47(3)
3.2 Properties of the Square Trigonometric Functions
50(1)
3.3 Period of the Square Trigonometric Functions in the Variable
51(2)
3.4 Fourier Series of the Square Trigonometric Functions
53(3)
3.5 Dynamic System Interpretation Of |x| + |y| = 1
56(3)
3.6 Hyperbolic Square Functions: Dynamics System Approach
59(5)
3.7 Periodic Hyperbolic Square Functions
64(3)
Notes and References
65(2)
4 Parabolic Trigonometric Functions
67(16)
4.1 Introduction
67(1)
4.2 H(x, y) = |y| + (1/2)x2 As a Dynamic System
67(5)
4.3 Geometric Analysis of |y| + (1/2)x2 = 1/2
72(1)
4.4 |y| -- (1/2)x2 = 1/2 As a Dynamic System
73(5)
4.5 Geometric Analysis of |y| -- (1/2)x2 = 1/2
78(5)
Notes and References
81(2)
5 Generalized Periodic Solutions of f(t)2 + g(t)2 = 1
83(16)
5.1 Introduction
83(1)
5.2 Generalized Cosine and Sine Functions
84(2)
5.3 Mathematical Structure of 9{t)
86(2)
5.4 An Example: A(t) = a1 sin (2π/T)t
88(1)
5.5 Differential Equation for f(t) AND g(t)
89(3)
5.6 Discussion
92(2)
5.7 Non-Periodic Solutions of f2(t) + g2(t) = 1
94(5)
Notes and References
97(2)
6 Resume of (Some) Previous Results on Generalized Trigonometric Functions
99(10)
6.1 Introduction
99(1)
6.2 Differential Equation Formulation
100(1)
6.3 Definition as Integral Forms
101(1)
6.4 Geometric Approach
102(1)
6.5 Symmetry Considerations and Consequences
103(4)
6.5.1 Symmetry Transformation and Consequences
103(1)
6.5.2 Hamiltonian Formulation
104(1)
6.5.3 Area of Enclosed Curve
105(1)
6.5.4 Period
106(1)
6.6 Summary
107(2)
Notes and References
107(2)
7 Generalized Trigonometric Functions: |y|P + |y|q = 1
109(10)
7.1 Introduction
109(1)
7.2 Methodology
109(2)
7.3 Summary
111(1)
7.4 Gallery of Particular Solutions
111(8)
8 Generalized Trigonometric Hyperbolic Functions: |y|p -- |x|q = 1
119(8)
8.1 Introduction
119(1)
8.2 Solutions
120(1)
8.3 Gallery of Special Solutions
120(7)
9 Applications and Advanced Topics
127(38)
9.1 Introduction
127(3)
9.2 Odd-Parity Systems and Their Fourier Representations
130(3)
9.3 Truly Nonlinear Oscillators
133(5)
9.3.1 Antisymmetric, Constant Force Oscillator
134(2)
9.3.2 Particle in a Box
136(1)
9.3.3 Restricted Doffing Equation
137(1)
9.4 Ateb Periodic Functions
138(2)
9.5 Exact Discretization of the Jacobi Elliptic Differential Equations
140(3)
9.5.1 Rescaled Duffing Equation
140(1)
9.5.2 Exact Difference Equation for CN
141(1)
9.5.3 Exact Difference Equation for SN
142(1)
9.6 Harmonic Balance: Direct Method
143(6)
9.6.1 Methodology
143(2)
9.6.2 x + x3 = 0
145(2)
9.6.3 x + x1/3 = 0
147(2)
9.7 Harmonic Balance: Rational Approximation
149(4)
9.7.1 Methodology
149(1)
9.7.2 x + x3 = 0
150(2)
9.7.3 x + x2 sgn(x) = 0
152(1)
9.8 Iteration Methods
153(5)
9.8.1 Direct Iteration Scheme: x + x3 = 0
155(2)
9.8.2 Extended Iteration: x + x3 = 0
157(1)
9.9 Discussion
158(7)
Notes and References
161(4)
1 Finale
165(6)
10.1 Goals
165(1)
10.2 Results
165(1)
10.3 Some Unresolved Topics and Issues
166(5)
Notes and References
168(3)
Appendix 171(14)
Bibliography 185(4)
Index 189
Ronald E. Mickens is the Distinguished Fuller E. Callaway Professor at Clark Atlanta University, Atlanta, GA, and is a Fellow of several professional organizations, including the American Physical Society. He has written or edited seventeen books and published more than 300 peer-reviewed research articles.
Permanent link: https://www.kriso.lv/db/9780429446238_pe.html
Keywords: