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http://functions.wolfram.com/14.03.20.0005.01
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f[x] Derivative[m][DiracDelta][x] ==
(-1)^m Sum[(-1)^k Binomial[m, k] Derivative[m - k][f][\[Xi]]
Derivative[k][DiracDelta][x], {k, 0, m}] /;
Element[m, Integers] && m >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["f", "[", "x", "]"]], RowBox[List[SuperscriptBox["DiracDelta", TagBox[RowBox[List["(", "m", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "x", "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", RowBox[List["m", "-", "k"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", "\[Xi]", "]"]], RowBox[List[SuperscriptBox["DiracDelta", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "x", "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "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> <mstyle scriptlevel='0'> <mi> f </mi> </mstyle> <mo> ⁢ </mo> <mstyle scriptlevel='0'> <mtext> </mtext> </mstyle> <mrow> <mo> ( </mo> <mstyle scriptlevel='0'> <mi> x </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> ⁢ </mo> <mstyle scriptlevel='0'> <mtext> </mtext> </mstyle> <mstyle scriptlevel='0'> <msup> <mi> δ </mi> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </msup> </mstyle> <mo> ⁢ </mo> <mstyle scriptlevel='0'> <mtext> </mtext> </mstyle> <mstyle scriptlevel='0'> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mi> ξ </mi> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <mstyle scriptlevel='0'> <mo> = </mo> </mstyle> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> ⁢ </mo> <mrow> <mstyle scriptlevel='0'> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> </mstyle> <mrow> <mstyle scriptlevel='0'> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mstyle> <mo> ⁢ </mo> <mstyle scriptlevel='0'> <mtext> </mtext> </mstyle> <mstyle scriptlevel='0'> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> m </mi> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> </mstyle> <mo> ⁢ </mo> <mstyle scriptlevel='0'> <mtext> </mtext> </mstyle> <mstyle scriptlevel='0'> <mrow> <msup> <mi> f </mi> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mi> ξ </mi> <mo> ) </mo> </mrow> </mrow> </mstyle> <mo> ⁢ </mo> <mstyle scriptlevel='0'> <mtext> </mtext> </mstyle> <mstyle scriptlevel='0'> <msup> <mi> δ </mi> <mrow> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </msup> </mstyle> <mo> ⁢ </mo> <mstyle scriptlevel='0'> <mtext> </mtext> </mstyle> <mrow> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <mi> x </mi> <mo> - </mo> <mi> ξ </mi> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Set </ci> <apply> <times /> <ci> f </ci> <ci> x </ci> <apply> <power /> <ci> δ </ci> <ci> m </ci> </apply> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ξ </ci> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <list> <list> <ci> m </ci> </list> <list> <ci> k </ci> </list> </list> <apply> <times /> <apply> <power /> <ci> f </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <ci> ξ </ci> </apply> <apply> <power /> <ci> δ </ci> <ci> k </ci> </apply> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ξ </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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