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http://functions.wolfram.com/01.27.10.0004.01
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ArcTanh[z] == z + z^3/(ContinueFraction[{(-(k - (-1)^k + 1)^2) z^2, 2 k + 3},
{k, 1, Infinity}] + 3) /;
!IntervalMemberQ[Interval[{-Infinity, -1}], z] &&
!IntervalMemberQ[Interval[{1, Infinity}], z]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTanh", "[", "z", "]"]], "\[Equal]", RowBox[List["z", "+", FractionBox[SuperscriptBox["z", "3"], RowBox[List[RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["k", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], "+", "1"]], ")"]], "2"]]], " ", SuperscriptBox["z", "2"]]], ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "3"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]], "+", "3"]]]]]]], "/;", "\[InvisibleSpace]", RowBox[List[RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]], "]"]], "\[And]", "\[InvisibleSpace]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", "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> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mrow> <mn> 3 </mn> <mo> + </mo> <msubsup> <mrow> <msub> <mi> Κ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 1 </mn> <mi> ∞ </mi> </msubsup> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mtext> ​ </mtext> <mrow> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mtext> ​ </mtext> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mrow> <mn> 3 </mn> <mo> + </mo> <msubsup> <mrow> <msub> <mi> Κ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 1 </mn> <mi> ∞ </mi> </msubsup> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mtext> ​ </mtext> <mrow> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mtext> ​ </mtext> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTanh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["z", "+", FractionBox[SuperscriptBox["z", "3"], RowBox[List[RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["k", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], "+", "1"]], ")"]], "2"]]], " ", SuperscriptBox["z", "2"]]], ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "3"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]], "+", "3"]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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