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http://functions.wolfram.com/14.01.07.0001.01
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UnitStep[x] == (1/(2 Pi I))
Limit[Integrate[E^(I t x)/(t - I \[CurlyEpsilon]),
{t, -Infinity, Infinity}], \[CurlyEpsilon] -> Plus[0]]
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Cell[BoxData[RowBox[List[RowBox[List["UnitStep", "[", "x", "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List["Limit", "[", RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "t", " ", "x"]]], RowBox[List["t", "-", RowBox[List["\[ImaginaryI]", " ", "\[CurlyEpsilon]"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> ε </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <msubsup> <mo> ∫ </mo> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mi> ∞ </mi> </msubsup> <mrow> <mfrac> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> t </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> </msup> <mrow> <mi> t </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ε </mi> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> UnitStep </ci> <ci> x </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <limit /> <bvar> <ci> ε </ci> </bvar> <condition> <apply> <tendsto /> <ci> ε </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> t </ci> <ci> x </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> ε </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["UnitStep", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["Limit", "[", RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "t", " ", "x"]]], RowBox[List["t", "-", RowBox[List["\[ImaginaryI]", " ", "\[CurlyEpsilon]"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
