LaplaceTransform[D[DiracDelta[t - a], {t, n}], t, x] == (x^n HeavisideTheta[a])/E^(a x) /; Element[n, Integers] && 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> <mrow> <msub> <mi> &#8466; </mi> <mi> t </mi> </msub> <mo> [ </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> n </mi> </msup> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> t </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> t </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> x </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> HeavisideTheta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[List[], Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LaplaceTransform </ci> <apply> <partialdiff /> <bvar> <ci> t </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> DiracDelta </ci> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <ci> t </ci> <ci> x </ci> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> x </ci> </apply> </apply> <apply> <power /> <ci> x </ci> <ci> n </ci> </apply> <apply> <ci> HeavisideTheta </ci> <ci> a </ci> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02