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ArcTanh




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.27.27.0022.01









  


  








Input Form





ArcTanh[Sqrt[1 - z^2]] == (1/Sqrt[1 - z^2]) (Sqrt[1 - 1/z^2] z ArcCsc[z] - (1/2) Pi Sqrt[z^2 - 1])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcTanh", "[", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]], " ", "z", " ", RowBox[List["ArcCsc", "[", "z", "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "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> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arctanh /> <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> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <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> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <arccsc /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <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> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTanh", "[", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z_", "2"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]], " ", "z", " ", RowBox[List["ArcCsc", "[", "z", "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]]


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





2001-10-29