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ArcSinh




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.25.27.0054.01









  


  








Input Form





ArcSinh[Sqrt[z^2 - 1]] == ArcCosh[z] + ((Pi I)/2) (Sqrt[z^2]/z - 1) /; Re[z] > 0 || Im[z] > 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSinh", "[", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["ArcCosh", "[", "z", "]"]], "+", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], RowBox[List["(", RowBox[List[FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], "-", "1"]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "||", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ">", "0"]]]]]]]]








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> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mi> z </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsinh /> <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> <apply> <plus /> <apply> <arccosh /> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["ArcCosh", "[", "z", "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], "-", "1"]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "||", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ">", "0"]]]]]]]]]]


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





2001-10-29