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1: 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
2: 4.37 Inverse Hyperbolic Functions
§4.37 Inverse Hyperbolic Functions
4.37.4 Arccsch z = Arcsinh ( 1 / z ) ,
4.37.7 arccsch z = arcsinh ( 1 / z ) ,
4.37.8 arcsech z = arccosh ( 1 / z ) .
4.37.9 arccoth z = arctanh ( 1 / z ) , z ± 1 .
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: 4.23 Inverse Trigonometric Functions
§4.23 Inverse Trigonometric Functions
4.23.6 Arccot z = Arctan ( 1 / z ) .
4.23.7 arccsc z = arcsin ( 1 / z ) ,
4.23.8 arcsec z = arccos ( 1 / z ) .
4.23.9 arccot z = arctan ( 1 / z ) , z ± i .
5: 18.28 Askey–Wilson Class
18.28.9 Q n ( 1 2 ( a q - y + a - 1 q y ) ; a , b | q - 1 ) = ( - 1 ) n b n q - 1 2 n ( n - 1 ) ( ( a b ) - 1 ; q ) n ϕ 1 3 ( q - n , q - y , a - 2 q y ( a b ) - 1 ; q , q n a b - 1 ) .
18.28.10 y = 0 ( 1 - q 2 y a - 2 ) ( a - 2 , ( a b ) - 1 ; q ) y ( 1 - a - 2 ) ( q , b q a - 1 ; q ) y ( b a - 1 ) y q y 2 Q n ( 1 2 ( a q - y + a - 1 q y ) ; a , b | q - 1 ) Q m ( 1 2 ( a q - y + a - 1 q y ) ; a , b | q - 1 ) = ( q a - 2 ; q ) ( b a - 1 q ; q ) ( q , ( a b ) - 1 ; q ) n ( a b ) n q - n 2 δ n , m .
18.28.18 h n ( sinh t | q ) = = 0 n q 1 2 ( + 1 ) ( q - n ; q ) ( q ; q ) e ( n - 2 ) t = e n t ϕ 1 1 ( q - n 0 ; q , - q e - 2 t ) = i - n H n ( i sinh t | q - 1 ) .
6: 19.2 Definitions
19.2.17 R C ( x , y ) = 1 2 0 d t t + x ( t + y ) ,
In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). When x and y are positive, R C ( x , y ) is an inverse circular function if x < y and an inverse hyperbolic function (or logarithm) if x > y :
19.2.18 R C ( x , y ) = 1 y - x arctan y - x x = 1 y - x arccos x / y , 0 x < y ,
19.2.19 R C ( x , y ) = 1 x - y arctanh x - y x = 1 x - y ln x + x - y y , 0 < y < x .
7: 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).
8: 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 . …
9: 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. …
10: 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 .