Hyperbolic Functions - Softcover

This compendium of hyperbolic trigonometry was first published as a chapter in Merriman and Woodward’s Higher Mathematics. There is reason to believe that it supplies a need, being adapted to two or three different types of readers. College students who have had elementary courses in trigonometry, analytic geometry, and differential and integral calculus, and who wish to know something of the hyperbolic trigonometry on account of its important and historic relations to each of those branches, will, it is hoped, find these relations presented in a simple and comprehensive way in the first half of the work. Readers who have some interest in imaginaries are then introduced to the more general trigonometry of the complex plane, where the circular and hyperbolic functions merge into one class of transcendent, the singly periodic functions, having either a real or a pure imaginary period. For those who also wish to view the subject in some of its practical relations, numerous applications have been selected so as to illustrate the various parts of the theory, and to show its use to the physicist and engineer, appropriate numerical tables being supplied for these purposes. With all these things in mind, much thought has been given to the mode of approaching the subject, and to the presentation of fundamental notions, and it is hoped that some improvements are discernible. For instance, it has been customary to define the hyperbolic functions in relation to a sector of the rectangular hyperbola, and to take the initial radius of the sector coincident with the principal radius of the curve; in the present work, these and similar restrictions are discarded in the interest of analogy and generality, with a gain in symmetry and simplicity, and the functions are defined as certain characteristic ratios belonging to any sector of any hyperbola. Such definitions, in connection with the fruitful notion of correspondence of points on conics, lead to simple and general proofs of the addition-theorems, from which easily follow the conversion-formulas, the derivatives, the Maclaurin expansions, and the exponential expressions. The proofs are so arranged as to apply equally to the circular functions, regarded as the characteristic ratios belonging to any elliptic sector. For those, however, who may wish to start with the exponential expressions as the definitions of the hyperbolic functions, the appropriate order of procedure is indicated on page 27, and a direct mode of bringing such exponential definitions into geometrical relation with the hyperbolic sector is shown in the Appendix CONTENTS: 1 Correspondence of Points on Conics 2 Areas of Corresponding Triangles 3 Areas of Corresponding Sectors 4 Charactersitic Ratios of Sectorial Measures 5 Ratios Expressed as Triangle-measures 6 Functional Relations for Ellipse 7 Functional Relations for Hyperbola 8 Relations Among Hyperbolic Functions 9 Variations of the Hyperbolic Functions 10 Anti-hyperbolic Functions 11 Functions of Sums and Difference 12 Conversion Formulas 13 Limiting Ratios 14 Derivatives of Hyperbolic Functions 15 Derivatives of Anti-hyperbolic Functions 16 Expansion of Hyperbolic Functions 17 Exponential Expressions 18 Expansion of Anti-functions 19 Logarithmic Expression of Anti-Functions 20 The Gudermanian Function 21 Circular Functions of Gudermanian 22 Gudermanian Angle 23 Derivatives of Gudermanian and Inverse 24 Series for Gudermanian and its Inverse 25 Graphs of Hyperbolic Functions 26 Elementary Integrals 27 Functions of Complex Numbers 28 Addition-Theorems for Complexes 29 Functions of Pure Imaginaries 30 Functions of x + iy in the Form X + iY 31 The Catenary 32 Catenary of Uniform Strength. 33 The Elastic Catenary 34 The Tractory 35 The Loxodrome 36 Combined Flexure and Tension 37 Alternating Currents 38 Miscellaneous Applications 39 Explanation of Tables 40 Appendix 40.1 Historical and Bibliographical 40.2 Exponential Expressions as Definitions