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1: 4.37 Inverse Hyperbolic Functions
§4.37 Inverse Hyperbolic Functions
Inverse Hyperbolic Sine
Inverse Hyperbolic Cosine
Inverse Hyperbolic Tangent
Other Inverse Functions
2: 4.23 Inverse Trigonometric Functions
§4.23 Inverse Trigonometric Functions
Inverse Sine
Inverse Cosine
Inverse Tangent
Other Inverse Functions
3: 22.15 Inverse Functions
§22.15 Inverse Functions
§22.15(i) Definitions
Each of these inverse functions is multivalued. The principal values satisfy …
4: 4.27 Sums
§4.27 Sums
For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).
5: 4.47 Approximations
§4.47 Approximations
§4.47(i) Chebyshev-Series Expansions
Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for ln , exp , sin , cos , tan , cot , arcsin , arctan , arcsinh . … Hart et al. (1968) give ln , exp , sin , cos , tan , cot , arcsin , arccos , arctan , sinh , cosh , tanh , arcsinh , arccosh . … Luke (1975, Chapter 3) supplies real and complex approximations for ln , exp , sin , cos , tan , arctan , arcsinh . …
6: 4.29 Graphics
§4.29(i) Real Arguments
See accompanying text
Figure 4.29.2: Principal values of arcsinh x and arccosh x . … Magnify
See accompanying text
Figure 4.29.4: Principal values of arctanh x and arccoth 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. …
7: 4.1 Special Notation
The main purpose of the present chapter is to extend these definitions and properties to complex arguments z . The main functions treated in this chapter are the logarithm ln z , Ln z ; the exponential exp z , e z ; the circular trigonometric (or just trigonometric) functions sin z , cos z , tan z , csc z , sec z , cot z ; the inverse trigonometric functions arcsin z , Arcsin z , etc. ; 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. Sometimes in the literature the meanings of ln and Ln are interchanged; similarly for arcsin z and Arcsin z , etc. … sin - 1 z for arcsin z and Sin - 1 z for Arcsin z .
8: 7.17 Inverse Error Functions
§7.17 Inverse Error Functions
§7.17(i) Notation
The inverses of the functions x = erf y , x = erfc y , y , are denoted by …
§7.17(ii) Power Series
§7.17(iii) Asymptotic Expansion of inverfc x for Small x
9: 4.24 Inverse Trigonometric Functions: Further Properties
§4.24 Inverse Trigonometric Functions: Further Properties
§4.24(i) Power Series
§4.24(ii) Derivatives
§4.24(iii) Addition Formulas
4.24.17 Arctan u ± Arccot v = Arctan ( u v ± 1 v u ) = Arccot ( v u u v ± 1 ) .
10: 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 ) .