Inverse hyperbolic cosine: Complex characteristics (formula 01.26.19.0007)

ArcCosh

$Mathematica Notation$

Elementary Functions

ArcCosh[z]

Complex characteristics

Conjugate value

http://functions.wolfram.com/01.26.19.0007.01

Input Form

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

Standard Form

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MathML Form

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</mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mroot> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mroot> </mrow> <mo> + </mo> <mi> x </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mroot> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mroot> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mroot> </mrow> <mo> + </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> OverBar </ci> <apply> <arccosh /> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <root /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <cos /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <arctan /> <apply> <plus /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> <ci> y </ci> </apply> <apply> <arctan /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <sin /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <arctan /> <apply> <plus /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> <ci> y </ci> </apply> <apply> <arctan /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <arctan /> <apply> <plus 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Rule Form

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

2001-10-29