Integrate[ParabolicCylinderD[\[Nu], t]^2, {t, 0, Infinity}] == (Sqrt[Pi]/(2^(3/2) Gamma[-\[Nu]])) (PolyGamma[(1 - \[Nu])/2] - PolyGamma[-(\[Nu]/2)])

Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], "2"], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SqrtBox["\[Pi]"], " "]], RowBox[List[SuperscriptBox["2", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]]]], ")"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <msup> <mrow> <msub> <semantics> <mi> D </mi> <annotation encoding='Mathematica'> TagBox[&quot;D&quot;, ParabolicCylinderD] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mtext> </mtext> </mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <power /> <apply> <ci> ParabolicCylinderD </ci> <ci> &#957; </ci> <ci> t </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <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[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]], "2"], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]]]], ")"]]]], RowBox[List[SuperscriptBox["2", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]]]]]]

2007-05-02