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variants of this functions
LaguerreL




Mathematica Notation

Traditional Notation









Polynomials > LaguerreL[n,z] > Operations > Orthogonality, completeness, and Fourier expansions





http://functions.wolfram.com/05.02.25.0004.01









  


  








Input Form





Integrate[(Exp[-(t/2)] LaguerreL[m, t]) (Exp[-(t/2)] LaguerreL[n, t]), {t, 0, Infinity}] == KroneckerDelta[n, m]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Exp", "[", RowBox[List["-", FractionBox["t", "2"]]], "]"]], RowBox[List["LaguerreL", "[", RowBox[List["m", ",", "t"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["Exp", "[", RowBox[List["-", FractionBox["t", "2"]]], "]"]], RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "t"]], "]"]]]], ")"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List["KroneckerDelta", "[", RowBox[List["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> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mi> t </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> L </mi> <mi> m </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mi> t </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> L </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> </msub> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> t </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> m </ci> <ci> t </ci> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> t </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> n </ci> <ci> t </ci> </apply> </apply> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["t_", "2"]]]], " ", RowBox[List["LaguerreL", "[", RowBox[List["m_", ",", "t_"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["t_", "2"]]]], " ", RowBox[List["LaguerreL", "[", RowBox[List["n_", ",", "t_"]], "]"]]]], ")"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["KroneckerDelta", "[", RowBox[List["n", ",", "m"]], "]"]]]]]]


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





2001-10-29