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variants of this functions
DiracDelta




Mathematica Notation

Traditional Notation









Generalized Functions > DiracDelta[x] > Limit representations





http://functions.wolfram.com/14.03.09.0010.01









  


  








Input Form





DiracDelta[x] == Limit[Abs[((1/\[CurlyEpsilon]) LaguerreL[n, (2 x)/\[CurlyEpsilon]])/ E^(x^2/\[CurlyEpsilon])], \[CurlyEpsilon] -> 0] /; Element[n, Integers] && n >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DiracDelta", "[", "x", "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List[FractionBox["1", "\[CurlyEpsilon]"], SuperscriptBox["\[ExponentialE]", RowBox[List["-", " ", FractionBox[RowBox[List[" ", SuperscriptBox["x", RowBox[List[" ", "2"]]]]], RowBox[List[" ", "\[CurlyEpsilon]"]]]]]], " ", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", " ", FractionBox[RowBox[List["2", " ", "x"]], RowBox[List[" ", "\[CurlyEpsilon]"]]]]], "]"]]]], "]"]], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", "0"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]








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> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> &#949; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mfrac> <mn> 1 </mn> <mi> &#949; </mi> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mi> &#949; </mi> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> L </mi> <mi> n </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mi> &#949; </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> DiracDelta </ci> <ci> x </ci> </apply> <apply> <limit /> <bvar> <ci> &#949; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#949; </ci> <cn type='integer'> 0 </cn> </apply> </condition> <apply> <abs /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> &#949; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> &#949; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> <apply> <power /> <ci> &#949; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DiracDelta", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["x", "2"], "\[CurlyEpsilon]"]]]], " ", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", FractionBox[RowBox[List["2", " ", "x"]], "\[CurlyEpsilon]"]]], "]"]]]], "\[CurlyEpsilon]"], "]"]], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", "0"]]]], "]"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2001-10-29