Subscript[x, n]^k D[DiracDelta[Subscript[x, 1], Subscript[x, 2], \[Ellipsis], Subscript[x, n]], Subscript[x, n]^m] == (((-1)^k m!)/(m - k)!) D[DiracDelta[Subscript[x, 1], Subscript[x, 2], \[Ellipsis], Subscript[x, n]], Subscript[x, n]^(m - k)] /; (Element[m, Integers] && m > 0) && (Element[k, Integers] && k >= 0) && k < m

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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> <msubsup> <mi> x </mi> <mi> n </mi> <mi> k </mi> </msubsup> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> m </mi> </msup> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> x </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> x </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> x </mi> <mi> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msubsup> <mi> x </mi> <mi> n </mi> <mi> m </mi> </msubsup> </mrow> </mfrac> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> x </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> x </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> x </mi> <mi> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msubsup> <mi> x </mi> <mi> n </mi> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &lt; </mo> <mi> m </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> n </ci> </apply> <ci> k </ci> </apply> <apply> <ci> D </ci> <apply> <ci> DiracDelta </ci> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> n </ci> </apply> </apply> <list> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> n </ci> </apply> <ci> m </ci> </list> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <factorial /> <ci> m </ci> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> D </ci> <apply> <ci> DiracDelta </ci> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> n </ci> </apply> </apply> <list> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> n </ci> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </list> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <in /> <ci> k </ci> <integers /> </apply> <apply> <lt /> <ci> k </ci> <ci> m </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29