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variants of this functions
HeavisideTheta




Mathematica Notation

Traditional Notation









Generalized Functions > HeavisideTheta[x] > Integral transforms > Inverse Fourier exp transforms





http://functions.wolfram.com/14.05.22.0002.01









  


  








Input Form





InverseFourierTransform[HeavisideTheta[t], t, x] == -(I/(Sqrt[2 Pi] x)) + Sqrt[Pi/2] DiracDelta[x]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["InverseFourierTransform", "[", RowBox[List[RowBox[List["HeavisideTheta", "[", "t", "]"]], ",", "t", ",", "x"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", "x"]]]]], "+", RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["DiracDelta", "[", "x", "]"]]]]]]]]]]








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> <msubsup> <mi> &#8497; </mi> <mi> t </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> [ </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> HeavisideTheta </ci> </annotation-xml> </semantics> <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> <mrow> <mo> - </mo> <mfrac> <mi> &#8520; </mi> <mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> InverseFourierTransform </ci> <apply> <ci> HeavisideTheta </ci> <ci> t </ci> </apply> <ci> t </ci> <ci> x </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> DiracDelta </ci> <ci> x </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseFourierTransform", "[", RowBox[List[RowBox[List["HeavisideTheta", "[", "t_", "]"]], ",", "t_", ",", "x_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", "x"]]]]], "+", RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["DiracDelta", "[", "x", "]"]]]]]]]]]]


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





2007-05-02