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http://functions.wolfram.com/03.21.09.0003.01
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SphericalBesselJ[\[Nu], z] == 2^(-1 - \[Nu]) Sqrt[Pi] z^\[Nu]
Limit[n^(-(1/2) - \[Nu]) LaguerreL[n, 1/2 + \[Nu], z^2/(4 n)],
n -> Infinity]
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Cell[BoxData[RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Nu]"], RowBox[List["Limit", "[", RowBox[List[RowBox[List[SuperscriptBox["n", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", FractionBox[SuperscriptBox["z", "2"], RowBox[List["4", " ", "n"]]]]], "]"]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]]]]
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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> <msub> <mi> j </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> ν </mi> </msup> <mo> ⁢ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <msup> <mi> n </mi> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mi> L </mi> <mi> n </mi> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <mo> ( </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> SphericalBesselJ </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <ci> ν </ci> </apply> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <power /> <ci> n </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> n </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["Limit", "[", RowBox[List[RowBox[List[SuperscriptBox["n", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", FractionBox[SuperscriptBox["z", "2"], RowBox[List["4", " ", "n"]]]]], "]"]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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