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http://functions.wolfram.com/14.03.21.0005.01
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Integrate[D[DiracDelta[t], {t, m}] D[DiracDelta[x - t], {t, n}],
{t, -Infinity, Infinity}] == D[DiracDelta[x], {x, m + n}] /;
Element[m, Integers] && m > 0 && Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[RowBox[List["D", "[", RowBox[List[RowBox[List["DiracDelta", "[", "t", "]"]], ",", RowBox[List["{", RowBox[List["t", ",", "m"]], "}"]]]], "]"]], RowBox[List["D", "[", RowBox[List[RowBox[List["DiracDelta", "[", RowBox[List["x", "-", "t"]], "]"]], ",", RowBox[List["{", RowBox[List["t", ",", "n"]], "}"]]]], "]"]], " ", RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List["D", "[", RowBox[List[RowBox[List["DiracDelta", "[", "x", "]"]], ",", RowBox[List["{", RowBox[List["x", ",", RowBox[List["m", "+", "n"]]]], "}"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "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> <msubsup> <mo> ∫ </mo> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> m </mi> </msup> <mrow> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> t </mi> <mi> m </mi> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <mrow> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> t </mi> <mi> n </mi> </msup> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo>  </mo> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> </msup> <mrow> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> x </mi> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> ∈ </mo> <msup> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> t </ci> <degree> <ci> m </ci> </degree> </bvar> <apply> <ci> DiracDelta </ci> <ci> t </ci> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> t </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> DiracDelta </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> x </ci> <degree> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </degree> </bvar> <apply> <ci> DiracDelta </ci> <ci> x </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["t_", ",", "m_"]], "}"]]]]], RowBox[List["DiracDelta", "[", "t_", "]"]]]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["t_", ",", "n_"]], "}"]]]]], RowBox[List["DiracDelta", "[", RowBox[List["x_", "-", "t_"]], "]"]]]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["x", ",", RowBox[List["m", "+", "n"]]]], "}"]]]]], RowBox[List["DiracDelta", "[", "x", "]"]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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