Hyperbolic Functions (illustrated)

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