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http://functions.wolfram.com/01.07.06.0065.01
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Cos[\[Lambda] x] HeavisideTheta[x] \[Proportional]
Sum[(-1)^k D[DiracDelta[x], {x, 2 k + 1}] \[Lambda]^(-2 k - 2),
{k, 0, Infinity}] /; (\[Lambda] -> Infinity)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Lambda]", " ", "x"]], "]"]], RowBox[List["HeavisideTheta", "[", "x", "]"]]]], "\[Proportional]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["D", "[", RowBox[List[RowBox[List["DiracDelta", "[", "x", "]"]], ",", RowBox[List["{", RowBox[List["x", ",", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]]], "}"]]]], "]"]], SuperscriptBox["\[Lambda]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "k"]], "-", "2"]]]]]]]]], "/;", RowBox[List["(", RowBox[List["\[Lambda]", "\[Rule]", "\[Infinity]"]], ")"]]]]]]
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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> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> HeavisideTheta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> ∝ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </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> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <times /> <apply> <cos /> <apply> <times /> <ci> λ </ci> <ci> x </ci> </apply> </apply> <apply> <ci> HeavisideTheta </ci> <ci> x </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <partialdiff /> <bvar> <ci> x </ci> <degree> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </degree> </bvar> <apply> <ci> DiracDelta </ci> <ci> x </ci> </apply> </apply> <apply> <power /> <ci> λ </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> λ </ci> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Lambda]_", " ", "x_"]], "]"]], " ", RowBox[List["HeavisideTheta", "[", "x_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["x", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "}"]]]]], RowBox[List["DiracDelta", "[", "x", "]"]]]], " ", SuperscriptBox["\[Lambda]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", "2"]]]]]]], "/;", RowBox[List["(", RowBox[List["\[Lambda]", "\[Rule]", "\[Infinity]"]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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