KelvinBer[z] == (1/2) Limit[LaguerreL[n, -((I z^2)/(4 n))] + LaguerreL[n, (I z^2)/(4 n)], n -> Infinity]

Cell[BoxData[RowBox[List[RowBox[List["KelvinBer", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], RowBox[List["Limit", "[", RowBox[List[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List["n", ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], RowBox[List["4", " ", "n"]]]]]]], "]"]], "+", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], RowBox[List["4", " ", "n"]]]]], "]"]]]], ",", RowBox[List["n", "\[Rule]", "Infinity"]]]], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> ber </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> L </mi> <mi> n </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msub> <mi> L </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> KelvinBer </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <plus /> <apply> <ci> LaguerreL </ci> <ci> n </ci> <apply> <times /> <imaginaryi /> <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> <ci> LaguerreL </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <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> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinBer", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Limit", "[", RowBox[List[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List["n", ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], RowBox[List["4", " ", "n"]]]]]]], "]"]], "+", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], RowBox[List["4", " ", "n"]]]]], "]"]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]]]]

2007-05-02