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variants of this functions
DiracDelta




Mathematica Notation

Traditional Notation









Generalized Functions > DiracDelta[x] > Limit representations





http://functions.wolfram.com/14.03.09.0012.01









  


  








Input Form





Derivative[n][DiracDelta][x] == Limit[((1/Sqrt[Pi \[CurlyEpsilon]]) (-(1/Sqrt[\[CurlyEpsilon]]))^n HermiteH[n, x/Sqrt[\[CurlyEpsilon]]])/E^(x^2/\[CurlyEpsilon]), \[CurlyEpsilon] -> Plus[0]] /; Element[n, Integers] && n > 0








Standard Form





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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> <msup> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <semantics> <mrow> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;n&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <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> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#949; </mi> </mrow> </msqrt> </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> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msqrt> <mi> &#949; </mi> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mfrac> <mi> x </mi> <msqrt> <mi> &#949; </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> x </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> DiracDelta </ci> <ci> x </ci> </apply> </apply> <apply> <limit /> <bvar> <ci> &#949; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#949; </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <pi /> <ci> &#949; </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <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> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> &#949; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> n </ci> </apply> <apply> <ci> HermiteH </ci> <ci> n </ci> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <power /> <ci> &#949; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29