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ArcSinh




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.25.27.0462.01









  


  








Input Form





ArcSinh[1/Sqrt[-1 - z^2]] == (Pi I)/2 - I ArcTan[z] /; 0 < Arg[z] < Pi/2 || (Element[I z, Reals] && I z < -1)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSinh", "[", FractionBox["1", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", "z", "]"]]]]]]]], "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "<", FractionBox["\[Pi]", "2"]]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "<", RowBox[List["-", "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> <mn> 1 </mn> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8744; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> &#8712; </mo> <mi> &#8477; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> &lt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsinh /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <arctan /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <arg /> <ci> z </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <ci> &#8477; </ci> </apply> <apply> <lt /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", FractionBox["1", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z_", "2"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", "z", "]"]]]]]], "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "<", FractionBox["\[Pi]", "2"]]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "<", RowBox[List["-", "1"]]]]]], ")"]]]]]]]]]]


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





2003-08-21