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inverse hyperbolic functions

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11: 4.45 Methods of Computation
Hyperbolic and Inverse Hyperbolic Functions
Similarly for the hyperbolic and inverse hyperbolic functions; compare (4.28.1)–(4.28.7), §4.37(iv), and (4.37.7)–(4.37.9). …
12: 19.17 Graphics
Because the R -function is homogeneous, there is no loss of generality in giving one variable the value 1 or 1 (as in Figure 19.3.2). …The case y = 1 corresponds to elementary functions. …
See accompanying text
Figure 19.17.8: R J ( 0 , y , 1 , p ) , 0 y 1 , 1 p 2 . …The function is asymptotic to 3 2 π / y p as p 0 + , and to ( 3 2 / p ) ln ( 16 / y ) as y 0 + . … Magnify 3D Help
13: 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 : …
19.2.19 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 .
14: 19.12 Asymptotic Approximations
19.12.6 R C ( x , y ) = π 2 y x y ( 1 + O ( x y ) ) , x / y 0 ,
19.12.7 R C ( x , y ) = 1 2 x ( ( 1 + y 2 x ) ln ( 4 x y ) y 2 x ) ( 1 + O ( y 2 / x 2 ) ) , y / x 0 .
15: 19.26 Addition Theorems
19.26.11 R C ( x + λ , y + λ ) + R C ( x + μ , y + μ ) = R C ( x , y ) ,
19.26.13 R C ( α 2 , α 2 θ ) + R C ( β 2 , β 2 θ ) = R C ( σ 2 , σ 2 θ ) , σ = ( α β + θ ) / ( α + β ) ,
19.26.14 ( p y ) R C ( x , p ) + ( q y ) R C ( x , q ) = ( η ξ ) R C ( ξ , η ) , x 0 , y 0 ; p , q { 0 } ,
19.26.22 R J ( x , y , z , p ) = 2 R J ( x + λ , y + λ , z + λ , p + λ ) + 3 R C ( α 2 , β 2 ) ,
19.26.25 R C ( x , y ) = 2 R C ( x + λ , y + λ ) , λ = y + 2 x y .
16: 4.32 Inequalities
4.32.4 arctan x 1 2 π tanh x , x 0 .
17: 19.11 Addition Theorems
19.11.6_5 R C ( γ δ , γ ) = 1 δ arctan ( δ sin θ sin ϕ sin ψ α 2 1 α 2 cos θ cos ϕ cos ψ ) .
18: 19.20 Special Cases
19.20.5 2 R G ( x , y , y ) = y R C ( x , y ) + x .
19.20.13 2 ( p x ) R J ( x , y , z , p ) = 3 R F ( x , y , z ) 3 x R C ( y z , p 2 ) , p = x ± ( y x ) ( z x ) ,
When the variables are real and distinct, the various cases of R J ( x , y , z , p ) are called circular (hyperbolic) cases if ( p x ) ( p y ) ( p z ) is positive (negative), because they typically occur in conjunction with inverse circular (hyperbolic) functions. …
19.20.20 R D ( x , y , y ) = 3 2 ( y x ) ( R C ( x , y ) x y ) , x y , y 0 ,
19.20.21 R D ( x , x , z ) = 3 z x ( R C ( z , x ) 1 z ) , x z , x z 0 .
19: 19.27 Asymptotic Approximations and Expansions
19.27.13 R J ( x , y , z , p ) = 3 2 z p ( ln ( 8 z a + g ) 2 R C ( 1 , p z ) + O ( ( a z + a p ) ln p a ) ) , max ( x , y ) / min ( z , p ) 0 .
19.27.14 R J ( x , y , z , p ) = 3 y z R C ( x , p ) 6 y z R G ( 0 , y , z ) + O ( x + 2 p y z ) , max ( x , p ) / min ( y , z ) 0 .
19.27.16 R J ( x , y , z , p ) = ( 3 / x ) R C ( ( h + p ) 2 , 2 ( b + h ) p ) + O ( 1 x 3 / 2 ln x b + h ) , max ( y , z , p ) / x 0 .
20: 14.15 Uniform Asymptotic Approximations
14.15.27 1 2 ζ ( ζ 2 α 2 ) 1 / 2 1 2 α 2 arccosh ( ζ α ) = ( 1 a 2 ) 1 / 2 arctanh ( 1 x ( x 2 a 2 1 a 2 ) 1 / 2 ) arccosh ( x a ) , a x < 1 , α ζ < ,
The inverse hyperbolic and trigonometric functions take their principal values (§§4.23(ii), 4.37(ii)). …
14.15.31 1 2 ζ ( ζ 2 + α 2 ) 1 / 2 + 1 2 α 2 arcsinh ( ζ α ) = ( 1 + a 2 ) 1 / 2 arctanh ( x ( 1 + a 2 x 2 + a 2 ) 1 / 2 ) arcsinh ( x a ) , 1 < x < 1 , < ζ < ,
(The inverse hyperbolic functions again take their principal values.) …