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ArcTanh




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.27.27.2115.01









  


  








Input Form





ArcTanh[1/Sqrt[1 + z]] == ((Sqrt[z] Sqrt[-1 - z])/(Sqrt[-z] Sqrt[1 + z])) ArcSinh[1/Sqrt[z]] + (Sqrt[(z + 1)/z] Sqrt[z/(z + 1)] - 1) ((Pi I)/2)








Standard Form





Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List["ArcTanh", "[", FractionBox["1", SqrtBox[RowBox[List["1", "+", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SqrtBox["z"], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", "z"]]]]], RowBox[List[SqrtBox[RowBox[List["-", "z"]]], SqrtBox[RowBox[List["1", "+", "z"]]]]]], RowBox[List["ArcSinh", "[", FractionBox["1", SqrtBox["z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[FractionBox[RowBox[List["z", "+", "1"]], "z"]], SqrtBox[FractionBox["z", RowBox[List["z", "+", "1"]]]]]], "-", "1"]], ")"]], FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]]]]]]]]]]








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> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mi> z </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arctanh /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <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> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arcsinh /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTanh", "[", FractionBox["1", SqrtBox[RowBox[List["1", "+", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["z"], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", "z"]]]]], ")"]], " ", RowBox[List["ArcSinh", "[", FractionBox["1", SqrtBox["z"]], "]"]]]], RowBox[List[SqrtBox[RowBox[List["-", "z"]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[FractionBox[RowBox[List["z", "+", "1"]], "z"]], " ", SqrtBox[FractionBox["z", RowBox[List["z", "+", "1"]]]]]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]]]]]]]


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





2003-08-21