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.
- 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
Table of Contents
1 TRIGONOMETRIC AND HYPERBOLIC SINE AND CO-SINE FUNCTIONS
2 ELLIPTIC FUNCTIONS
3 SQUARE FUNCTIONS
4 PARABOLIC TRIGONOMETRIC FUNCTIONS
5 GENERALIZED PERIODIC SOLUTIONS OF f(t)2+g(t)2 = 1
6 RESUME OF (SOME) PREVIOUS RESULTS ON GENERALIZED TRIGONOMETRIC FUNCTIONS 99
7 GENERALIZED TRIGONOMETRIC FUNCTIONS: |y|p +|x|q = 1
8 GENERALIZED TRIGONOMETRIC HYPERBOLIC FUNCTIONS: |y|p − |x|q = 1
9 APPLICATIONS AND ADVANCED TOPICS
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.
"Differing from the standard approach, this book argues the benefits of creating and analyzing generalized trigonometric functions using methodologies involving geometrical phase-space and dynamical systems. This is a mouthful, but Mickens (Clark Atlanta Univ.) takes on his subject with vigor. After a brief overview of the standard approaches and results, Mickens discusses square and parabolic forms of periodic and hyperbolic functions, as well as their generalized forms. From this basic understanding, he uses special functional equations to define a ‘new class’ of generalized trigonometric and hyperbolic functions, which he uses to model important dynamical systems."
—J. Johnson, Emeritus Professor, Western Washington University