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ArcSinh




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSinh[z] > Representations through equivalent functions > With related functions > Involving cot-1 > Involving sinh-1(2 z/z2-1) > Involving sinh-1(2 z/z2-1) and cot-1(i z)





http://functions.wolfram.com/01.25.27.0659.01









  


  








Input Form





ArcSinh[(2 z)/(z^2 - 1)] == 2 I ArcCot[I z] /; Abs[z] > 1








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSinh", "[", FractionBox[RowBox[List["2", " ", "z"]], RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]], "]"]], "\[Equal]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcCot", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mtext> </mtext> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <arccot /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", FractionBox[RowBox[List["2", " ", "z_"]], RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcCot", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]]]]]]]]


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





2003-08-21