Sign[Log[z]] == ((1/2) Log[Re[z]^2 + Im[z]^2] + I ArcTan[Re[z], Im[z]])/ Sqrt[ArcTan[Re[z], Im[z]]^2 + (1/4) Log[Re[z]^2 + Im[z]^2]^2]

Cell[BoxData[RowBox[List[RowBox[List["Sign", "[", RowBox[List["Log", "[", "z", "]"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List[FractionBox["1", "2"], RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["Re", "[", "z", "]"]], "2"], "+", SuperscriptBox[RowBox[List["Im", "[", "z", "]"]], "2"]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]], "2"], "+", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["Re", "[", "z", "]"]], "2"], "+", SuperscriptBox[RowBox[List["Im", "[", "z", "]"]], "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> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <msup> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <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> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Sign </ci> <apply> <ln /> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <arctan /> <apply> <real /> <ci> z </ci> </apply> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ln /> <apply> <plus /> <apply> <power /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <arctan /> <apply> <real /> <ci> z </ci> </apply> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <power /> <apply> <ln /> <apply> <plus /> <apply> <power /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </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["Log", "[", "z_", "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["Re", "[", "z", "]"]], "2"], "+", SuperscriptBox[RowBox[List["Im", "[", "z", "]"]], "2"]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ",", RowBox[List["Im", "[", "z", "]"]]]], "]"]], "2"], "+", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["Re", "[", "z", "]"]], "2"], "+", SuperscriptBox[RowBox[List["Im", "[", "z", "]"]], "2"]]], "]"]], "2"]]]]]]]]]]]

2007-05-02