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1: 4.37 Inverse Hyperbolic Functions
§4.37 Inverse Hyperbolic Functions
§4.37(i) General Definitions
§4.37(ii) Principal Values
Figure 4.37.1: z -plane. …
§4.37(iv) Logarithmic Forms
2: 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. …
3: 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 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. …
4: 4.46 Tables
§4.46 Tables
5: 4.47 Approximations
§4.47 Approximations
§4.47(i) Chebyshev-Series Expansions
6: 19.10 Relations to Other Functions
7: 4.39 Continued Fractions
§4.39 Continued Fractions
4.39.2 arcsinh z 1 + z 2 = z 1 + 1 2 z 2 3 + 1 2 z 2 5 + 3 4 z 2 7 + 3 4 z 2 9 + ,
4.39.3 arctanh z = z 1 - z 2 3 - 4 z 2 5 - 9 z 2 7 - ,
For these and other continued fractions involving inverse hyperbolic functions see Lorentzen and Waadeland (1992, pp. 569–571). …
8: 4.38 Inverse Hyperbolic Functions: Further Properties
§4.38 Inverse Hyperbolic Functions: Further Properties
§4.38(i) Power Series
§4.38(ii) Derivatives
§4.38(iii) Addition Formulas
4.38.19 Arctanh u ± Arccoth v = Arctanh ( u v ± 1 v ± u ) = Arccoth ( v ± u u v ± 1 ) .
9: 4.40 Integrals
§4.40 Integrals
§4.40(iv) Inverse Hyperbolic Functions
4.40.11 arcsinh x d x = x arcsinh x - ( 1 + x 2 ) 1 / 2 .
4.40.14 arccsch x d x = x arccsch x + arcsinh x , 0 < x < ,
4.40.15 arcsech x d x = x arcsech x + arcsin x , 0 < x < 1 ,
10: 4.23 Inverse Trigonometric Functions