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http://functions.wolfram.com/01.14.27.0856.01
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ArcTan[z] == 2 ArcCsc[Sqrt[(2 Sqrt[1 + z^2])/(Sqrt[1 + z^2] + z)]] - Pi/2 /;
!IntervalMemberQ[Interval[{-Infinity, -1}], I z]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["2", RowBox[List["ArcCsc", "[", SqrtBox[FractionBox[RowBox[List["2", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]], RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]], "+", "z"]]]], "]"]]]], "-", FractionBox["\[Pi]", "2"]]]]], "/;", 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"]]]], "]"]], "]"]]]]]]
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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> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mi> z </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> /; </mo> <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> </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> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mi> z </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> /; </mo> <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> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTan", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["2", " ", RowBox[List["ArcCsc", "[", SqrtBox[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]], RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]], "+", "z"]]]], "]"]]]], "-", FractionBox["\[Pi]", "2"]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
