Generalized Trigonometric and Hyperbolic Functions

Ronald E. Mickens

• Verlag: CRC Press
• Jahr: 2019
• ISBN:
  9780429821080
• Auflage: 1
• Thema: Mathematics
• Thema: Mathematics General

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

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• Clear, direct construction of a new set of generalized trigonometric and hyperbolic functions

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• Presentation of why x2+y2 = 1, and related expressions, may be interpreted in three distinct ways

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• All the constructions, proofs, and derivations can be readily followed and understood by students, researchers, and professionals in the natural and mathematical sciences

Chapter 1

TRIGONOMETRIC AND HYPERBOLIC SINE AND COSINE FUNCTIONS

1.1 INTRODUCTION

Historically, the trigonometric sine and cosine functions had their genesis in the study of properties of the magnitude of line segments contained within the perimeter of a circle. This chapter examines these functions from several different perspectives and ends with the introduction of a new set of functions which we denote the θ-periodic functions.

Section 1.2 starts with the standard definition of the sine/cosine functions based on the geometric properties of the circle. Section 1.3 is based on an analytic definition of sine/cosine which follows from the Euler relation

$e^{i\theta }=\cos \theta +i\sin \theta .$

(1.1)

Another interesting and important way of examining the properties of sine/cosine functions is to consider them solutions to a dynamic system modeling the simple harmonic oscillator. These results are presented in Section 1.4. The next three sections are devoted to the ass...