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http://functions.wolfram.com/05.08.21.0008.01
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Integrate[(t^\[Lambda] LaguerreL[m, \[Lambda], t] LaguerreL[n, \[Lambda], t])/
E^t, {t, 0, Infinity}] == (Gamma[\[Lambda] + n + 1]/n!)
KroneckerDelta[n, m] /; Re[\[Lambda]] > -1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t", "\[Lambda]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "t"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List["m", ",", "\[Lambda]", ",", "t"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "\[Lambda]", ",", "t"]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "n", "+", "1"]], "]"]], RowBox[List["n", "!"]]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["n", ",", "m"]], "]"]]]]]], "/;", " ", RowBox[List[RowBox[List["Re", "[", "\[Lambda]", "]"]], ">", RowBox[List["-", "1"]]]]]]]]
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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> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <msup> <mi> t </mi> <mi> λ </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mi> t </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mi> L </mi> <mi> m </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> L </mi> <mi> n </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> </msub> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> λ </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <ci> λ </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> m </ci> <ci> λ </ci> <ci> t </ci> </apply> <apply> <ci> LaguerreL </ci> <ci> n </ci> <ci> λ </ci> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> λ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", "\[Lambda]_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "t_"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List["m_", ",", "\[Lambda]_", ",", "t_"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List["n_", ",", "\[Lambda]_", ",", "t_"]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "n", "+", "1"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["n", ",", "m"]], "]"]]]], RowBox[List["n", "!"]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Lambda]", "]"]], ">", RowBox[List["-", "1"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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