FourierTransform[Sign[t], t, x] == I Sqrt[2/Pi] (1/x)

Cell[BoxData[RowBox[List[RowBox[List["FourierTransform", "[", RowBox[List[RowBox[List["Sign", "[", "t", "]"]], ",", "t", ",", "x"]], "]"]], "\[Equal]", RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox["2", "\[Pi]"]], FractionBox["1", "x"]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> &#8497; </mi> <mi> t </mi> </msub> <mo> [ </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <msqrt> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mtext> </mtext> </mrow> <mi> x </mi> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> FourierTransform </ci> <apply> <ci> Sign </ci> <ci> t </ci> </apply> <ci> t </ci> <ci> x </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FourierTransform", "[", RowBox[List[RowBox[List["Sign", "[", "t_", "]"]], ",", "t_", ",", "x_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox["2", "\[Pi]"]]]], "x"]]]]]

2007-05-02