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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/01.27.27.1442.01









  


  








Input Form





ArcTanh[Sqrt[1 - z^2]] == ((z Sqrt[-1 + z^2])/Sqrt[z^2 - z^4]) ArcCsc[z] + (Pi Sqrt[1 - z^2])/(2 Sqrt[z^2 - 1])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcTanh", "[", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["z", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", SuperscriptBox["z", "4"]]]]], RowBox[List["ArcCsc", "[", "z", "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], RowBox[List["2", 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> <mrow> <mfrac> <mrow> <mi> z </mi> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </msqrt> </mfrac> <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> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </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> <plus /> <apply> <times /> <apply> <times /> <ci> z </ci> <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> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccsc /> <ci> z </ci> </apply> </apply> <apply> <times /> <pi /> <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> <times /> <cn type='integer'> 2 </cn> <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> <cn type='integer'> -1 </cn> </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]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List["ArcCsc", "[", "z", "]"]]]], SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", SuperscriptBox["z", "4"]]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]]]]]]]]


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





2003-08-21