Sign[IntegerPart[x + I y]] == (IntegerPart[x] + I IntegerPart[y])/ Sqrt[IntegerPart[x]^2 + IntegerPart[y]^2]

Cell[BoxData[RowBox[List[RowBox[List["Sign", "[", RowBox[List["IntegerPart", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["IntegerPart", "[", "x", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["IntegerPart", "[", "y", "]"]]]]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["IntegerPart", "[", "x", "]"]], "2"], "+", SuperscriptBox[RowBox[List["IntegerPart", "[", "y", "]"]], "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> int </mi> <mo> &#8289; </mo> <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> <mi> int </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> int </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <msup> <mrow> <mi> int </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> int </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Sign </ci> <apply> <ci> int </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> int </ci> <ci> x </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <ci> int </ci> <ci> y </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> int </ci> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> int </ci> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </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["IntegerPart", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["IntegerPart", "[", "x", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["IntegerPart", "[", "y", "]"]]]]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["IntegerPart", "[", "x", "]"]], "2"], "+", SuperscriptBox[RowBox[List["IntegerPart", "[", "y", "]"]], "2"]]]]]]]]]

2007-05-02