DiracDelta[f[x]] == Sum[DiracDelta[x - Subscript[x, j]]/ Abs[Derivative[1][f][Subscript[x, j]]], {j, 1, n}] /; f[Subscript[x, j]] == 0 && Derivative[1][f][Subscript[x, j]] != 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> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mfrac> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <msub> <mi> x </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <msup> <mi> f </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <msub> <mi> x </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> x </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msup> <mi> f </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <msub> <mi> x </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> DiracDelta </ci> <apply> <ci> f </ci> <ci> x </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> DiracDelta </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> j </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <abs /> <apply> <ci> D </ci> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> j </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> j </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> j </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <neq /> <apply> <ci> D </ci> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> j </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> j </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29