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http://functions.wolfram.com/14.03.09.0001.01
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DiracDelta[x] == Limit[UnitStep[x + 1] UnitStep[1 - x] (\[CurlyEpsilon]/2)
Abs[x]^(\[CurlyEpsilon] - 1), \[CurlyEpsilon] -> Plus[0]]
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Cell[BoxData[RowBox[List[RowBox[List["DiracDelta", "[", "x", "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["x", "+", "1"]], "]"]], RowBox[List["UnitStep", "[", RowBox[List["1", "-", "x"]], "]"]], FractionBox["\[CurlyEpsilon]", "2"], " ", SuperscriptBox[RowBox[List["Abs", "[", "x", "]"]], RowBox[List["\[CurlyEpsilon]", "-", "1"]]]]], ",", 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> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <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> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> ε </mi> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> x </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mrow> <mi> ε </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> DiracDelta </ci> <ci> x </ci> </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> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ci> UnitStep </ci> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <ci> ε </ci> <apply> <power /> <apply> <abs /> <ci> x </ci> </apply> <apply> <plus /> <ci> ε </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DiracDelta", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["UnitStep", "[", RowBox[List["x", "+", "1"]], "]"]], " ", RowBox[List["UnitStep", "[", RowBox[List["1", "-", "x"]], "]"]], " ", "\[CurlyEpsilon]", " ", SuperscriptBox[RowBox[List["Abs", "[", "x", "]"]], RowBox[List["\[CurlyEpsilon]", "-", "1"]]]]], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
