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http://functions.wolfram.com/07.41.23.0003.01
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Sum[(w^k/k!) ParabolicCylinderD[\[Nu] + 2 k, z], {k, 0, Infinity}] ==
(1/(1 + 2 w)^((\[Nu] + 1)/2)) E^((w z^2)/(2 (2 w + 1)))
ParabolicCylinderD[\[Nu], z/Sqrt[2 w + 1]] /; Abs[w] < 1/2
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox["w", "k"], RowBox[List["k", "!"]]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List[RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "k"]]]], ",", "z"]], "]"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "w"]]]], ")"]], FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"]]], SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["w", " ", SuperscriptBox["z", "2"]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "w"]], "+", "1"]], ")"]]]]]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", FractionBox["z", SqrtBox[RowBox[List[RowBox[List["2", " ", "w"]], "+", "1"]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", FractionBox["1", "2"]]]]]]]
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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> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msup> <mi> w </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> D </mi> <annotation encoding='Mathematica'> TagBox["D", ParabolicCylinderD] </annotation> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> ν </mi> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo>  </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> w </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mi> w </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> w </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> D </mi> <annotation encoding='Mathematica'> TagBox["D", ParabolicCylinderD] </annotation> </semantics> <mi> ν </mi> </msub> <mo> ( </mo> <mfrac> <mi> z </mi> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> w </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> w </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> w </ci> <ci> k </ci> </apply> <apply> <ci> ParabolicCylinderD </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <ci> ν </ci> </apply> <ci> z </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> w </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> ParabolicCylinderD </ci> <ci> ν </ci> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> w </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["w_", "k_"], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List[RowBox[List["\[Nu]_", "+", RowBox[List["2", " ", "k_"]]]], ",", "z_"]], "]"]]]], RowBox[List["k_", "!"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["w", " ", SuperscriptBox["z", "2"]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "w"]], "+", "1"]], ")"]]]]]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", FractionBox["z", SqrtBox[RowBox[List[RowBox[List["2", " ", "w"]], "+", "1"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "w"]]]], ")"]], FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"]]], "/;", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", FractionBox["1", "2"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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