Sign[ArcTan[x + I y]] == (ArcTan[1 - y, x] - ArcTan[1 + y, -x] - (1/2) I (Log[x^2 + (-1 + y)^2] - Log[x^2 + (1 + y)^2]))/ Sqrt[(ArcTan[1 - y, x] - ArcTan[1 + y, -x])^2 + (1/4) (Log[x^2 + (-1 + y)^2] - Log[x^2 + (1 + y)^2])^2]

Cell[BoxData[RowBox[List[RowBox[List["Sign", "[", RowBox[List["ArcTan", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", "y"]], ",", "x"]], "]"]], "-", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "y"]], ",", RowBox[List["-", "x"]]]], "]"]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "y"]], ")"]], "2"]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "y"]], ")"]], "2"]]], "]"]]]], ")"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", "y"]], ",", "x"]], "]"]], "-", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "y"]], ",", RowBox[List["-", "x"]]]], "]"]]]], ")"]], "2"], "+", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "y"]], ")"]], "2"]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "y"]], ")"]], "2"]]], "]"]]]], ")"]], "2"]]]]], ")"]]]], ")"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> y </mi> </mrow> <mo> , </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> y </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> x </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> y </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> y </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> y </mi> </mrow> <mo> , </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> y </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> x </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> y </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> y </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Sign </ci> <apply> <arctan /> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <arctan /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctan /> <apply> <plus /> <ci> y </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <plus /> <apply> <ln /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> y </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <arctan /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctan /> <apply> <plus /> <ci> y </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <power /> <apply> <plus /> <apply> <ln /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> y </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sign", "[", RowBox[List["ArcTan", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", "y"]], ",", "x"]], "]"]], "-", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "y"]], ",", RowBox[List["-", "x"]]]], "]"]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "y"]], ")"]], "2"]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "y"]], ")"]], "2"]]], "]"]]]], ")"]]]]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", "y"]], ",", "x"]], "]"]], "-", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "y"]], ",", RowBox[List["-", "x"]]]], "]"]]]], ")"]], "2"], "+", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "y"]], ")"]], "2"]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "y"]], ")"]], "2"]]], "]"]]]], ")"]], "2"]]]]]]]]]]]

2007-05-02