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ArcCosh






Mathematica Notation

Traditional Notation









Elementary Functions > ArcCosh[z] > Transformations > Related transformations > Differences involving the direct function > Involving sinh-1(z)





http://functions.wolfram.com/01.26.16.0207.01









  


  










Input Form





ArcCosh[x] - ArcSinh[y] == (I Sqrt[(-1)^Floor[1/2 - Arg[I Sqrt[x - 1] Sqrt[x + 1] Sqrt[y^2 + 1] - I x y]/Pi] (x Sqrt[y^2 + 1] - Sqrt[x - 1] Sqrt[x + 1] y) + 1] ArcCosh[(-1)^Floor[1/2 - Arg[I Sqrt[x - 1] Sqrt[x + 1] Sqrt[y^2 + 1] - I x y]/Pi] (Sqrt[x - 1] Sqrt[x + 1] y - x Sqrt[y^2 + 1])])/ Sqrt[(-(-1)^Floor[1/2 - Arg[I Sqrt[x - 1] Sqrt[x + 1] Sqrt[y^2 + 1] - I x y]/Pi]) (x Sqrt[y^2 + 1] - Sqrt[x - 1] Sqrt[x + 1] y) - 1] - (1/2) I Pi ((1/2) (1 - (-1)^(-Floor[-(Arg[1 - x]/(2 Pi))])) (2 (-1 + (-1)^Floor[1/2 - Arg[Sqrt[1 - x^2] Sqrt[y^2 + 1] - I x y]/Pi]) Floor[(Arg[I x + Sqrt[1 - x^2]] + Arg[Sqrt[y^2 + 1] - y])/(2 Pi)] + (-1)^Floor[1/2 - Arg[Sqrt[1 - x^2] Sqrt[y^2 + 1] - I x y]/Pi] + 2 (1 + (-1)^Floor[1/2 - Arg[Sqrt[1 - x^2] Sqrt[y^2 + 1] - I x y]/Pi]) Floor[1/2 - (Arg[I x + Sqrt[1 - x^2]] + Arg[Sqrt[y^2 + 1] - y])/ (2 Pi)]) - (1/2) (1 + (-1)^(-Floor[-(Arg[1 - x]/(2 Pi))])) (2 (-1 + (-1)^Floor[1/2 - Arg[I x y + Sqrt[1 - x^2] Sqrt[y^2 + 1]]/Pi]) Floor[(Arg[I x + Sqrt[1 - x^2]] + Arg[y + Sqrt[y^2 + 1]])/(2 Pi)] + (-1)^Floor[1/2 - Arg[I x y + Sqrt[1 - x^2] Sqrt[y^2 + 1]]/Pi] + 2 (1 + (-1)^Floor[1/2 - Arg[I x y + Sqrt[1 - x^2] Sqrt[y^2 + 1]]/Pi]) Floor[1/2 - (Arg[I x + Sqrt[1 - x^2]] + Arg[y + Sqrt[y^2 + 1]])/ (2 Pi)])) - (I Pi)/2










Standard Form





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MathML Form







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&#8520; </mi> <mo> &#8290; </mo> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> - </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> 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<plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> <ci> y </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <arg /> <apply> <plus /> <ci> y </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <arg /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> 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Rule Form





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





2007-05-02





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