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Power






Mathematica Notation

Traditional Notation









Elementary Functions > Power[z,a] > Integral transforms > Fourier exp transforms





http://functions.wolfram.com/01.02.22.0004.01









  


  










Input Form





FourierTransform[UnitStep[t]/t^n, t, x] == (x^(n - 1) I^(n - 1) (2 PolyGamma[n] - Log[x^2] + Pi I Sign[x]))/ (2 Sqrt[2 Pi] (n - 1)!) /; Element[n, Integers] && n > 0










Standard Form





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MathML Form







<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> &#8497; </mi> <mi> t </mi> </msub> <mo> [ </mo> <mfrac> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <msup> <mi> t </mi> <mi> n </mi> </msup> </mfrac> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mi> x </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8520; </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> FourierTransform </ci> <apply> <times /> <apply> <ci> UnitStep </ci> <ci> t </ci> </apply> <apply> <power /> <apply> <power /> <ci> t </ci> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> t </ci> <ci> x </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> x </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <imaginaryi /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyGamma </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <pi /> <apply> <ci> Sign </ci> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FourierTransform", "[", RowBox[List[FractionBox[RowBox[List["UnitStep", "[", "t_", "]"]], SuperscriptBox["t_", "n_"]], ",", "t_", ",", "x_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["x", RowBox[List["n", "-", "1"]]], " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["PolyGamma", "[", "n", "]"]]]], "-", RowBox[List["Log", "[", SuperscriptBox["x", "2"], "]"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Sign", "[", "x", "]"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29