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ArcSinh




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.25.27.1836.01









  


  








Input Form





ArcSinh[(2 Sqrt[z^2 + 1])/z^2] == 2 ArcCsch[z] /; Abs[z] >= Sqrt[2] && Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi/2]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSinh", "[", FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]]]], SuperscriptBox["z", "2"]], "]"]], "\[Equal]", RowBox[List["2", RowBox[List["ArcCsch", "[", "z", "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[GreaterEqual]", SqrtBox["2"]]], "\[And]", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]








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> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <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> &#8805; </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> &lt; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsinh /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arccsch /> <ci> z </ci> </apply> </apply> </apply> <apply> <and /> <apply> <geq /> <apply> <abs /> <ci> z </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Inequality </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <lt /> <apply> <arg /> <ci> z </ci> </apply> <leq /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "+", "1"]]]]], SuperscriptBox["z_", "2"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["2", " ", RowBox[List["ArcCsch", "[", "z", "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[GreaterEqual]", SqrtBox["2"]]], "&&", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]]]


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





2003-08-21