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http://functions.wolfram.com/05.06.25.0001.01
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Limit[((1/n^a) JacobiP[n, a, b, Cos[z/n]])/(2/z)^a, n -> Infinity] ==
BesselJ[a, z]
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Cell[BoxData[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["1", " "]], SuperscriptBox["n", "a"]], SuperscriptBox[RowBox[List["(", FractionBox["2", "z"], ")"]], RowBox[List["-", "a"]]], RowBox[List["JacobiP", "[", RowBox[List["n", ",", "a", ",", "b", ",", RowBox[List["Cos", "[", FractionBox["z", "n"], "]"]]]], "]"]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "\[Equal]", RowBox[List["BesselJ", "[", RowBox[List["a", ",", "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> n </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> n </mi> <mi> a </mi> </msup> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mi> P </mi> <mi> n </mi> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msub> <mi> J </mi> <mi> a </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> n </ci> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <ci> JacobiP </ci> <ci> n </ci> <ci> a </ci> <ci> b </ci> <apply> <cos /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> a </ci> <ci> z </ci> </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[SuperscriptBox[RowBox[List["(", FractionBox["2", "z_"], ")"]], RowBox[List["-", "a_"]]], " ", RowBox[List["JacobiP", "[", RowBox[List["n_", ",", "a_", ",", "b_", ",", RowBox[List["Cos", "[", FractionBox["z_", "n_"], "]"]]]], "]"]]]], SuperscriptBox["n_", "a_"]], ",", RowBox[List["n_", "\[Rule]", "\[Infinity]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["BesselJ", "[", RowBox[List["a", ",", "z"]], "]"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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