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variants of this functions
DiracDelta




Mathematica Notation

Traditional Notation









Generalized Functions > DiracDelta[x] > Differentiation > Symbolic differentiation





http://functions.wolfram.com/14.03.20.0004.01









  


  








Input Form





x^n Derivative[m][DiracDelta][x] == (-1)^n (m!/(m - n)!) Derivative[m - n][DiracDelta][x] /; Element[m, Integers] && Element[n, Integers] && Inequality[0, LessEqual, n, Less, m]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["x", "n"], RowBox[List[SuperscriptBox["DiracDelta", TagBox[RowBox[List["(", "m", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "x", "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], FractionBox[RowBox[List["m", "!"]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], "!"]]], RowBox[List[SuperscriptBox["DiracDelta", TagBox[RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", "x", "]"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["0", "\[LessEqual]", "n", "<", "m"]]]]]]]]








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> <mi> x </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> &#948; </mi> <semantics> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;m&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> &#948; </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;n&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> n </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 /> <ci> x </ci> <ci> n </ci> </apply> <apply> <partialdiff /> <bvar> <ci> x </ci> <degree> <ci> m </ci> </degree> </bvar> <apply> <ci> &#948; </ci> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </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> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> x </ci> <degree> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </degree> </bvar> <apply> <ci> &#948; </ci> <ci> x </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <ci> Inequality </ci> <cn type='integer'> 0 </cn> <leq /> <ci> n </ci> <lt /> <ci> m </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["x_", "n_"], " ", RowBox[List[SuperscriptBox["DiracDelta", TagBox[RowBox[List["(", "m", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "x_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["m", "!"]], " ", RowBox[List[SuperscriptBox["DiracDelta", TagBox[RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", "x", "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], "!"]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["0", "\[LessEqual]", "n", "<", "m"]]]]]]]]]]


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





2001-10-29