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ArcCosh




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.26.27.1554.01









  


  








Input Form





ArcCosh[z] == Pi I + ArcTanh[Sqrt[1 - z^2]/Sqrt[-z^2]] /; Pi/2 <= Arg[z] <= Pi








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCosh", "[", "z", "]"]], "\[Equal]", " ", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]]]], "/;", RowBox[List[FractionBox["\[Pi]", "2"], "\[LessEqual]", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]]]]]]








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> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> &#8804; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mi> &#960; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arccosh /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <pi /> <imaginaryi /> </apply> <apply> <arctanh /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <leq /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arg /> <ci> z </ci> </apply> <pi /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCosh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]], "/;", RowBox[List[FractionBox["\[Pi]", "2"], "\[LessEqual]", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]]]]]]]]


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





2003-08-21