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http://functions.wolfram.com/01.14.10.0002.01
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ArcTan[z] == z/(1 + ContinueFraction[{k^2 z^2, 2 k + 1},
{k, 1, Infinity}]) /;
!IntervalMemberQ[Interval[{-Infinity, -1}], I z] &&
!IntervalMemberQ[Interval[{1, Infinity}], I z]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", "z", "]"]], "\[Equal]", FractionBox["z", RowBox[List["1", "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox["k", "2"], " ", SuperscriptBox["z", "2"]]], ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]]]]]]], "/;", "\[InvisibleSpace]", RowBox[List[RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], "]"]], "\[And]", "\[InvisibleSpace]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "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> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mfrac> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> + </mo> <msubsup> <mrow> <msub> <mi> Κ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msup> <mi> k </mi> <mn> 2 </mn> </msup> <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> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 1 </mn> <mi> ∞ </mi> </msubsup> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mtext> ​ </mtext> <mi> z </mi> </mrow> <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> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mtext> ​ </mtext> <mi> z </mi> </mrow> <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> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mfrac> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> + </mo> <msubsup> <mrow> <msub> <mi> Κ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msup> <mi> k </mi> <mn> 2 </mn> </msup> <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> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 1 </mn> <mi> ∞ </mi> </msubsup> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mtext> ​ </mtext> <mi> z </mi> </mrow> <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> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mtext> ​ </mtext> <mi> z </mi> </mrow> <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["ArcTan", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["z", RowBox[List["1", "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox["k", "2"], " ", SuperscriptBox["z", "2"]]], ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
