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ArcTanh




Mathematica Notation

Traditional Notation









Elementary Functions > ArcTanh[z] > Representations through equivalent functions > With related functions > Involving cosh-1 > Involving tanh-1(z1/2/(z-1)1/2) > Involving tanh-1(z1/2/(z-1)1/2) and cosh-1(z1/2)





http://functions.wolfram.com/01.27.27.2496.01









  


  








Input Form





ArcTanh[Sqrt[z]/Sqrt[z - 1]] == ArcCosh[Sqrt[z]] + (Pi I)/2 /; Im[z] < 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTanh", "[", FractionBox[SqrtBox["z"], SqrtBox[RowBox[List["z", "-", "1"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["ArcCosh", "[", SqrtBox["z"], "]"]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]]]], "/;", 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> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mi> z </mi> </msqrt> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arctanh /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <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> <arccosh /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox["z_"], SqrtBox[RowBox[List["z_", "-", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["ArcCosh", "[", SqrtBox["z"], "]"]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], "/;", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "<", "0"]]]]]]]]


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





2003-08-21