Integrate[(LegendreP[n, m, (1 - x)/(1 + x)] BesselJ[m, y Sqrt[x]])/ (1 + x)^(3/2), {x, 0, Infinity}] == (((-1)^n/(n + 1/2)) (2 y)^m LaguerreL[n - m, 2 m, 2 y])/E^y /; Element[n, Integers] && n >= 0 && Element[m, Integers] && 0 <= m <= n && Element[y, Reals] && y >= 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "m", ",", FractionBox[RowBox[List["1", "-", "x"]], RowBox[List["1", "+", "x"]]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["m", ",", RowBox[List["y", " ", SqrtBox["x"]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], RowBox[List["3", "/", "2"]]]], RowBox[List["\[DifferentialD]", "x"]]]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["n", "+", RowBox[List["1", "/", "2"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "y"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "y"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["n", "-", "m"]], ",", RowBox[List["2", " ", "m"]], ",", RowBox[List["2", " ", "y"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["0", "\[LessEqual]", "m", "\[LessEqual]", "n"]], "\[And]", RowBox[List["y", "\[Element]", "Reals"]], "\[And]", RowBox[List["y", "\[GreaterEqual]", "0"]]]]]]]]

<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> &#8747; </mo> <mn> 0 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mfrac> <mrow> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> x </mi> </mrow> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mi> m </mi> </msub> <mo> ( </mo> <mrow> <mi> y </mi> <mo> &#8290; </mo> <msqrt> <mi> x </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> x </mi> </mrow> </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> <mtext> </mtext> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> y </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mi> L </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msubsup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> m </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> y </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> y </mi> <mo> &#8805; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> x </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> m </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> m </ci> <apply> <times /> <ci> y </ci> <apply> <power /> <ci> x </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> </apply> <ci> m </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> <apply> <ci> LaguerreL </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> m </ci> <ci> n </ci> </apply> <apply> <in /> <ci> y </ci> <reals /> </apply> <apply> <geq /> <ci> y </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "m_", ",", FractionBox[RowBox[List["1", "-", "x_"]], RowBox[List["1", "+", "x_"]]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["m_", ",", RowBox[List["y_", " ", SqrtBox["x_"]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x_"]], ")"]], RowBox[List["3", "/", "2"]]]], RowBox[List["\[DifferentialD]", "x_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "y"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "y"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["n", "-", "m"]], ",", RowBox[List["2", " ", "m"]], ",", RowBox[List["2", " ", "y"]]]], "]"]]]], RowBox[List["n", "+", FractionBox["1", "2"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["0", "\[LessEqual]", "m", "\[LessEqual]", "n"]], "&&", RowBox[List["y", "\[Element]", "Reals"]], "&&", RowBox[List["y", "\[GreaterEqual]", "0"]]]]]]]]]]

2002-12-18