Limit[(((-(1/2))^n E^(z^2/(8 n)) Sqrt[n])/n!) ParabolicCylinderD[2 n, z/(Sqrt[2] Sqrt[n])], n -> Infinity] == (1/Sqrt[Pi]) Cos[z] /; Element[n, Integers] && n >= 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "n"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], RowBox[List["8", " ", "n"]]]], " ", SqrtBox["n"], " "]], RowBox[List["n", "!"]]], RowBox[List["ParabolicCylinderD", "[", RowBox[List[RowBox[List["2", " ", "n"]], ",", FractionBox["z", RowBox[List[SqrtBox["2"], " ", SqrtBox["n"]]]]]], "]"]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox["\[Pi]"]], RowBox[List["Cos", "[", "z", "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <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> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <msqrt> <mi> n </mi> </msqrt> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> D </mi> <annotation encoding='Mathematica'> TagBox[&quot;D&quot;, ParabolicCylinderD] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msub> <mo> ( </mo> <mfrac> <mi> z </mi> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> n </mi> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> &#63449; </mo> <mfrac> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> n </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> ParabolicCylinderD </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <cos /> <ci> z </ci> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "n_"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z_", "2"], RowBox[List["8", " ", "n_"]]]], " ", SqrtBox["n_"]]], ")"]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List[RowBox[List["2", " ", "n_"]], ",", FractionBox["z_", RowBox[List[SqrtBox["2"], " ", SqrtBox["n_"]]]]]], "]"]]]], RowBox[List["n_", "!"]]], ",", RowBox[List["n_", "\[Rule]", "\[Infinity]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["Cos", "[", "z", "]"]], SqrtBox["\[Pi]"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

2007-05-02