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http://functions.wolfram.com/01.27.06.0064.01
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ArcTanh[z]^2 == -(Pi^2/4) + ((Pi z)/Sqrt[-z^2]) ArcTanh[1/z] +
ArcTanh[1/z]^2 /; !IntervalMemberQ[Interval[{-1, 1}], z]
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Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcTanh", "[", "z", "]"]], "2"], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["\[Pi]", "2"], "4"]]], "+", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "z"]], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]], RowBox[List["ArcTanh", "[", FractionBox["1", "z"], "]"]]]], "+", SuperscriptBox[RowBox[List["ArcTanh", "[", FractionBox["1", "z"], "]"]], "2"]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msup> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["ArcTanh", "[", "z_", "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["\[Pi]", "2"], "4"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", "z"]], ")"]], " ", RowBox[List["ArcTanh", "[", FractionBox["1", "z"], "]"]]]], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]], "+", SuperscriptBox[RowBox[List["ArcTanh", "[", FractionBox["1", "z"], "]"]], "2"]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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