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http://functions.wolfram.com/14.03.02.0001.01
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DiracDelta[x] == (1/Pi) Limit[\[CurlyEpsilon]/(x^2 + \[CurlyEpsilon]^2),
\[CurlyEpsilon] -> 0] /; Element[x, Reals]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DiracDelta", "[", "x", "]"]], "\[Equal]", RowBox[List[FractionBox["1", "\[Pi]"], RowBox[List["Limit", "[", " ", RowBox[List[FractionBox["\[CurlyEpsilon]", RowBox[List[" ", RowBox[List[SuperscriptBox["x", RowBox[List[" ", "2"]]], " ", "+", " ", SuperscriptBox["\[CurlyEpsilon]", RowBox[List[" ", "2"]]]]]]]], ",", " ", RowBox[List["\[CurlyEpsilon]", " ", "\[Rule]", " ", "0"]]]], "]"]]]]]], "/;", " ", RowBox[List["x", "\[Element]", "Reals"]]]]]]
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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> <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> <mfrac> <mn> 1 </mn> <mi> π </mi> </mfrac> <mo> ⁢ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> ε </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mfrac> <mi> ε </mi> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> ε </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> x </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> DiracDelta </ci> <ci> x </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <limit /> <bvar> <ci> ε </ci> </bvar> <condition> <apply> <tendsto /> <ci> ε </ci> <cn type='integer'> 0 </cn> </apply> </condition> <apply> <times /> <ci> ε </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> ε </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DiracDelta", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["Limit", "[", RowBox[List[FractionBox["\[CurlyEpsilon]", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["\[CurlyEpsilon]", "2"]]]], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", "0"]]]], "]"]], "\[Pi]"], "/;", RowBox[List["x", "\[Element]", "Reals"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
