Subscript[\[Eta], i] Subscript[\[Omega], j] - Subscript[\[Eta], j] Subscript[\[Omega], i] == Sign[Re[(-I) (Subscript[\[Omega], j]/Subscript[\[Omega], i])]] ((Pi I)/2) /; Element[{i, j}, {1, 2, 3}] && i != j

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[Eta]", "i"], " ", SubscriptBox["\[Omega]", "j"]]], "-", RowBox[List[SubscriptBox["\[Eta]", "j"], " ", SubscriptBox["\[Omega]", "i"]]]]], "\[Equal]", RowBox[List[RowBox[List["Sign", "[", RowBox[List["Re", "[", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", FractionBox[SubscriptBox["\[Omega]", "j"], SubscriptBox["\[Omega]", "i"]]]], "]"]], "]"]], " ", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["i", ",", "j"]], "}"]], "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2", ",", "3"]], "}"]]]], "\[And]", RowBox[List["i", "\[NotEqual]", "j"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msub> <mi> &#951; </mi> <mi> i </mi> </msub> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <msub> <mi> &#951; </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mi> i </mi> </msub> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mfrac> <mrow> <mtext> </mtext> <msub> <mi> &#969; </mi> <mi> j </mi> </msub> </mrow> <msub> <mi> &#969; </mi> <mi> i </mi> </msub> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> </mrow> <mo> } </mo> </mrow> <mo> &#8712; </mo> <mrow> <mo> { </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> i </mi> <mo> &#8800; </mo> <mi> j </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <ci> i </ci> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <ci> j </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <ci> i </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Sign </ci> <apply> <real /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <ci> j </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <ci> i </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <list> <ci> i </ci> <ci> j </ci> </list> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> <cn type='integer'> 3 </cn> </list> </apply> <apply> <neq /> <ci> i </ci> <ci> j </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SubscriptBox["\[Eta]_", "i_"], " ", SubscriptBox["\[Omega]_", "j"]]], "-", RowBox[List[SubscriptBox["\[Eta]_", "j"], " ", SubscriptBox["\[Omega]_", "i_"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Sign", "[", RowBox[List["Re", "[", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SubscriptBox["\[Omega]", "j"]]], SubscriptBox["\[Omega]\[Omega]", "i"]]]], "]"]], "]"]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["i", ",", "j"]], "}"]], "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2", ",", "3"]], "}"]]]], "&&", RowBox[List["i", "\[NotEqual]", "j"]]]]]]]]]]

2001-10-29