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http://functions.wolfram.com/07.41.21.0014.01
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Integrate[(t^(\[Alpha] - 1) ParabolicCylinderD[\[Nu], t])/E^(a t^2),
{t, 0, Infinity}] == ((2^((\[Nu] - \[Alpha])/2) Sqrt[Pi] Gamma[\[Alpha]])/
Gamma[(1 + \[Alpha] - \[Nu])/2]) Hypergeometric2F1[\[Alpha]/2,
(\[Alpha] + 1)/2, (1 + \[Alpha] - \[Nu])/2, 1/2 - 2 a] /;
(Re[\[Alpha]] > 0 && Re[4 a + 1] > 0) ||
(0 < Re[\[Alpha]] < -Re[\[Nu]] && Re[4 a + 1] = 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", InterpretationBox["\[Infinity]", DirectedInfinity[1]]], " ", RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["t", "2"]]]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Alpha]", "]"]], " "]], RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Alpha]", "-", "\[Nu]"]], "2"], "]"]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["\[Alpha]", "2"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], ",", FractionBox[RowBox[List["1", "+", "\[Alpha]", "-", "\[Nu]"]], "2"], " ", ",", RowBox[List[FractionBox["1", "2"], " ", "-", RowBox[List["2", "a"]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List[RowBox[List["4", " ", "a"]], "+", "1"]], "]"]], ">", "0"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["0", "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["-", RowBox[List["Re", "[", "\[Nu]", "]"]]]]]], "&&", RowBox[List["Re", "[", RowBox[List[RowBox[List["4", " ", "a"]], "+", "1"]], "]"]]]], "=", "0"]], ")"]]]]]]]]
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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> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <msup> <mi> t </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ⁢ </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> D </mi> <annotation encoding='Mathematica'> TagBox["D", ParabolicCylinderD] </annotation> </semantics> <mi> ν </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo>  </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mfrac> <mrow> <mi> ν </mi> <mo> - </mo> <mi> α </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> α </mi> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> α </mi> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mrow> <mi> α </mi> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["\[Alpha]", "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["\[Alpha]", "-", "\[Nu]", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List[FractionBox["1", "2"], "-", RowBox[List["2", " ", "a"]]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∨ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 0 </mn> <mo> < </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> = </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> ParabolicCylinderD </ci> <ci> ν </ci> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <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> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <apply> <times /> <apply> <plus /> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> α </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <ci> Set </ci> <apply> <and /> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <real /> <ci> α </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> ν </ci> </apply> </apply> </apply> <apply> <real /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a_"]], " ", SuperscriptBox["t_", "2"]]]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", FractionBox[RowBox[List["\[Nu]", "-", "\[Alpha]"]], "2"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Alpha]", "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["\[Alpha]", "2"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "-", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], "-", RowBox[List["2", " ", "a"]]]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "-", "\[Nu]"]], ")"]]]], "]"]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List[RowBox[List["4", " ", "a"]], "+", "1"]], "]"]], ">", "0"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["0", "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["-", RowBox[List["Re", "[", "\[Nu]", "]"]]]]]], "&&", RowBox[List["Re", "[", RowBox[List[RowBox[List["4", " ", "a"]], "+", "1"]], "]"]]]], "=", "0"]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
