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1: 4.37 Inverse Hyperbolic Functions
§4.37 Inverse Hyperbolic Functions
The principal values of the inverse hyperbolic cosecant, hyperbolic secant, and hyperbolic tangent are given by …
Inverse Hyperbolic Sine
Inverse Hyperbolic Cosine
Inverse Hyperbolic Tangent
2: 36.2 Catastrophes and Canonical Integrals
Normal Forms for Umbilic Catastrophes with Codimension K = 3
36.2.3 Φ ( H ) ( s , t ; 𝐱 ) = s 3 + t 3 + z s t + y t + x s , 𝐱 = { x , y , z } ,
(hyperbolic umbilic).
Canonical Integrals
36.2.27 Ψ ( H ) ( x , y , z ) = Ψ ( H ) ( y , x , z ) .
3: 4.35 Identities
§4.35 Identities
§4.35(i) Addition Formulas
§4.35(ii) Squares and Products
§4.35(iii) Multiples of the Argument
§4.35(iv) Real and Imaginary Parts; Moduli
4: 4.29 Graphics
§4.29(i) Real Arguments
See accompanying text
Figure 4.29.6: Principal values of arccsch x and arcsech x . … Magnify
§4.29(ii) Complex Arguments
The conformal mapping w = sinh z is obtainable from Figure 4.15.7 by rotating both the w -plane and the z -plane through an angle 1 2 π , compare (4.28.8). The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …
5: 4.28 Definitions and Periodicity
§4.28 Definitions and Periodicity
Relations to Trigonometric Functions
As a consequence, many properties of the hyperbolic functions follow immediately from the corresponding properties of the trigonometric functions.
Periodicity and Zeros
The functions sinh z and cosh z have period 2 π i , and tanh z has period π i . …
6: 4.1 Special Notation
k , m , n integers.
The main purpose of the present chapter is to extend these definitions and properties to complex arguments z . … ; the hyperbolic trigonometric (or just hyperbolic) functions sinh z , cosh z , tanh z , csch z , sech z , coth z ; the inverse hyperbolic functions arcsinh z , Arcsinh z , etc. …
7: 4.32 Inequalities
§4.32 Inequalities
4.32.1 cosh x ( sinh x x ) 3 ,
4.32.2 sin x cos x < tanh x < x , x > 0 ,
4.32.3 | cosh x cosh y | | x y | sinh x sinh y , x > 0 , y > 0 ,
For these and other inequalities involving hyperbolic functions see Mitrinović (1964, pp. 61, 76, 159) and Mitrinović (1970, p. 270).
8: 4.31 Special Values and Limits
§4.31 Special Values and Limits
Table 4.31.1: Hyperbolic functions: values at multiples of 1 2 π i .
z 0 1 2 π i π i 3 2 π i
4.31.1 lim z 0 sinh z z = 1 ,
4.31.2 lim z 0 tanh z z = 1 ,
4.31.3 lim z 0 cosh z 1 z 2 = 1 2 .
9: 4.41 Sums
§4.41 Sums
For sums of hyperbolic functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §43), Prudnikov et al. (1986a, §5.3), and Zucker (1979).
10: 4.34 Derivatives and Differential Equations
§4.34 Derivatives and Differential Equations
4.34.4 d d z csch z = csch z coth z ,
4.34.5 d d z sech z = sech z tanh z ,
With a 0 , the general solutions of the differential equations …