Inverse hyperbolic sine: Complex characteristics (formula 01.25.19.0005)

ArcSinh

$Mathematica Notation$

Elementary Functions

ArcSinh[z]

Complex characteristics

Absolute value

http://functions.wolfram.com/01.25.19.0005.01

Input Form

Abs[ArcSinh[x + I y]] == Sqrt[ArcTan[x + (4 x^2 y^2 + (1 + x^2 - y^2)^2)^(1/4) Cos[(1/2) ArcTan[1 + x^2 - y^2, 2 x y]], y + (4 x^2 y^2 + (1 + x^2 - y^2)^2)^(1/4) Sin[(1/2) ArcTan[1 + x^2 - y^2, 2 x y]]]^2 + Log[Sqrt[(x + (4 x^2 y^2 + (1 + x^2 - y^2)^2)^(1/4) Cos[(1/2) ArcTan[1 + x^2 - y^2, 2 x y]])^2 + (y + (4 x^2 y^2 + (1 + x^2 - y^2)^2)^(1/4) Sin[(1/2) ArcTan[1 + x^2 - y^2, 2 x y]])^2]]^2]

Standard Form

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MathML Form

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<mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <abs /> <apply> <arcsinh /> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <root /> <apply> <plus /> <apply> <power /> <apply> <arctan /> <apply> <plus /> <apply> <times /> <apply> <cos /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <arctan /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y 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Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29