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http://functions.wolfram.com/07.03.25.0002.01
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Limit[(1/\[Nu]^\[Lambda]) LaguerreL[\[Nu], \[Lambda], -(z^2/(4 \[Nu]))],
\[Nu] -> Infinity] == (2^\[Lambda] BesselI[\[Lambda], z])/z^\[Lambda]
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Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox["\[Nu]", "\[Lambda]"]], RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], RowBox[List["4", " ", "\[Nu]"]]]]]]], "]"]]]], ",", RowBox[List["\[Nu]", "\[Rule]", "\[Infinity]"]]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", "\[Lambda]"], " ", SuperscriptBox["z", RowBox[List["-", "\[Lambda]"]]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Lambda]", ",", "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> <munder> <mi> lim </mi> <mrow> <mi> ν </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> ν </mi> <mi> λ </mi> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mi> L </mi> <mi> ν </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mn> 2 </mn> <mi> λ </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> λ </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mi> λ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <limit /> <bvar> <ci> ν </ci> </bvar> <condition> <apply> <tendsto /> <ci> ν </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> ν </ci> <ci> λ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> ν </ci> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> <apply> <ci> BesselI </ci> <ci> λ </ci> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", RowBox[List["-", FractionBox[SuperscriptBox["z_", "2"], RowBox[List["4", " ", "\[Nu]_"]]]]]]], "]"]], SuperscriptBox["\[Nu]_", "\[Lambda]_"]], ",", RowBox[List["\[Nu]_", "\[Rule]", "\[Infinity]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", "\[Lambda]"], " ", SuperscriptBox["z", RowBox[List["-", "\[Lambda]"]]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Lambda]", ",", "z"]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
