ArcTan[x, y] \[Proportional] (I/2) (1/(x Sqrt[1/x^2]) - 1) (Log[-y] - Log[y]) + ((Pi x)/(2 y)) Sqrt[y^2/x^2] - x/y + O[x^3/y^3] /; Element[y, Reals]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox["\[ImaginaryI]", "2"], RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["x", SqrtBox[FractionBox["1", SuperscriptBox["x", "2"]]]]]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "y"]], "]"]], "-", RowBox[List["Log", "[", "y", "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "x"]], RowBox[List["2", "y"]]], SqrtBox[FractionBox[SuperscriptBox["y", "2"], SuperscriptBox["x", "2"]]]]], "-", FractionBox["x", "y"], "+", RowBox[List["O", "[", FractionBox[SuperscriptBox["x", "3"], SuperscriptBox["y", "3"]], "]"]]]]]], "/;", RowBox[List["Element", "[", RowBox[List["y", ",", "Reals"]], "]"]]]]]]

<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> &#8733; </mo> <mrow> <mrow> <mfrac> <mi> &#8520; </mi> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> x </mi> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </msqrt> </mrow> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </msqrt> <mo> &#8290; </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mfrac> </mrow> <mo> - </mo> <mfrac> <mi> x </mi> <mi> y </mi> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </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> <mi> y </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </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 /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <pi /> <ci> x </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> </apply> <cn type='integer'> -1 </cn> </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> <in /> <ci> y </ci> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTan", "[", RowBox[List["x_", ",", "y_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["x", " ", SqrtBox[FractionBox["1", SuperscriptBox["x", "2"]]]]]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "y"]], "]"]], "-", RowBox[List["Log", "[", "y", "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", "x"]], ")"]], " ", SqrtBox[FractionBox[SuperscriptBox["y", "2"], SuperscriptBox["x", "2"]]]]], RowBox[List["2", " ", "y"]]], "-", FractionBox["x", "y"], "+", RowBox[List["O", "[", FractionBox[SuperscriptBox["x", "3"], SuperscriptBox["y", "3"]], "]"]]]], "/;", RowBox[List["y", "\[Element]", "Reals"]]]]]]]]

2001-10-29