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http://functions.wolfram.com/07.02.21.0004.01
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Integrate[z^(\[Nu] - 2) LaguerreL[\[Nu], z], z] ==
z^(-1 + \[Nu]) Gamma[\[Nu] - 1] HypergeometricPFQRegularized[
{-\[Nu], -1 + \[Nu]}, {1, \[Nu]}, z]
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Nu]", "-", "2"]]], RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "-", "1"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Nu]"]], "}"]], ",", "z"]], "]"]]]]]]]]
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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> <mo> ∫ </mo> <mrow> <msup> <mi> z </mi> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> L </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mi> z </mi> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> , </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> ν </mi> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Nu]"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "-", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Nu]", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> </list> <list> <cn type='integer'> 1 </cn> <ci> ν </ci> </list> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Nu]_", "-", "2"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "-", "1"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Nu]"]], "}"]], ",", "z"]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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