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11: 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 ) .
12: 4.45 Methods of Computation
Inverse Trigonometric Functions
The function arctan x can always be computed from its ascending power series after preliminary transformations to reduce the size of x . …
4.45.10 arctan x = 2 n arctan x n .
4.45.13 arctan x = 16 arctan x 4 = 1.46563 .
For the remaining inverse trigonometric functions, we may use the identities provided by the fourth row of Table 4.16.3. …
13: 4.25 Continued Fractions
§4.25 Continued Fractions
4.25.3 arcsin 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 ,
4.25.4 arctan z = z 1 + z 2 3 + 4 z 2 5 + 9 z 2 7 + 16 z 2 9 + ,
4.25.5 e 2 a arctan ( 1 / z ) = 1 + 2 a z a + a 2 + 1 3 z + a 2 + 4 5 z + a 2 + 9 7 z + ,
See Lorentzen and Waadeland (1992, pp. 560–571) for other continued fractions involving inverse trigonometric functions. …
14: 4.26 Integrals
4.26.5 sec x d x = gd 1 ( x ) , 1 2 π < x < 1 2 π .
§4.26(iv) Inverse Trigonometric Functions
4.26.14 arcsin x d x = x arcsin x + ( 1 x 2 ) 1 / 2 , 1 < x < 1 ,
4.26.15 arccos x d x = x arccos x ( 1 x 2 ) 1 / 2 , 1 < x < 1 .
Extensive compendia of indefinite and definite integrals of trigonometric and inverse trigonometric functions include Apelblat (1983, pp. 48–109), Bierens de Haan (1939), Gradshteyn and Ryzhik (2000, Chapters 2–4), Gröbner and Hofreiter (1949, pp. 116–139), Gröbner and Hofreiter (1950, pp. 94–160), and Prudnikov et al. (1986a, §§1.5, 1.7, 2.5, 2.7).
15: 4.40 Integrals
4.40.11 arcsinh x d x = x arcsinh x ( 1 + x 2 ) 1 / 2 .
4.40.12 arccosh x d x = x arccosh x ( x 2 1 ) 1 / 2 , 1 < x < ,
4.40.13 arctanh x d x = x arctanh x + 1 2 ln ( 1 x 2 ) , 1 < x < 1 ,
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 ,
16: 4.38 Inverse Hyperbolic Functions: Further Properties
4.38.9 d d z arcsinh z = ( 1 + z 2 ) 1 / 2 .
4.38.11 d d z arctanh z = 1 1 z 2 .
4.38.13 d d z arcsech z = 1 z ( 1 z 2 ) 1 / 2 .
4.38.18 Arcsinh u ± Arccosh v = Arcsinh ( u v ± ( ( 1 + u 2 ) ( v 2 1 ) ) 1 / 2 ) = Arccosh ( v ( 1 + u 2 ) 1 / 2 ± u ( v 2 1 ) 1 / 2 ) ,
4.38.19 Arctanh u ± Arccoth v = Arctanh ( u v ± 1 v ± u ) = Arccoth ( v ± u u v ± 1 ) .
17: 28.26 Asymptotic Approximations for Large q
28.26.3 ϕ = 2 h sinh z ( m + 1 2 ) arctan ( sinh z ) .
18: 19.30 Lengths of Plane Curves
19: 9.8 Modulus and Phase
9.8.4 θ ( x ) = arctan ( Ai ( x ) / Bi ( x ) ) .
9.8.8 ϕ ( x ) = arctan ( Ai ( x ) / Bi ( x ) ) .
9.8.11 θ ( x ) = 2 3 π + arctan ( Y 1 / 3 ( ξ ) / J 1 / 3 ( ξ ) ) ,
9.8.12 ϕ ( x ) = 1 3 π + arctan ( Y 2 / 3 ( ξ ) / J 2 / 3 ( ξ ) ) .
20: 19.2 Definitions
19.2.11_5 el1 ( x , k c ) = 0 arctan x 1 cos 2 θ + k c 2 sin 2 θ d θ ,
In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). …
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 .
19.2.20 R C ( x , y ) = x x y R C ( x y , y ) = 1 x y arctanh x x y = 1 x y ln x + x y y , y < 0 x .