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http://functions.wolfram.com/01.15.06.0004.02
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ArcTan[x, y] \[Proportional] (-I) Log[(I y)/Sqrt[y^2]] - x/y + O[x^3/y^3] /;
!Element[y, Reals] && !Element[I y, Reals]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["\[ImaginaryI]", " ", "y"]], SqrtBox[SuperscriptBox["y", "2"]]], "]"]]]], "-", FractionBox["x", "y"], "+", RowBox[List["O", "[", FractionBox[SuperscriptBox["x", "3"], SuperscriptBox["y", "3"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["y", ",", "Reals"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], ",", "Reals"]], "]"]], "]"]]]]]]]]
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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> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <msqrt> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mi> x </mi> <mi> y </mi> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <msup> <mi> x </mi> <mn> 3 </mn> </msup> <msup> <mi> y </mi> <mn> 3 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> y </mi> <mo> ∉ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> ∉ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <arctan /> <ci> x </ci> <ci> y </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <ln /> <apply> <times /> <imaginaryi /> <ci> y </ci> <apply> <power /> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <apply> <power /> <ci> x </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <notin /> <ci> y </ci> <reals /> </apply> <apply> <notin /> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> <reals /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTan", "[", RowBox[List["x_", ",", "y_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["\[ImaginaryI]", " ", "y"]], SqrtBox[SuperscriptBox["y", "2"]]], "]"]]]], "-", FractionBox["x", "y"], "+", RowBox[List["O", "[", FractionBox[SuperscriptBox["x", "3"], SuperscriptBox["y", "3"]], "]"]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["y", "\[Element]", "Reals"]]]], "&&", RowBox[List["!", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "\[Element]", "Reals"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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