
|

|

|

|

|
|

|

|

|

|

|
|

|

|

|

|

|
|

|

|

|

|
|

|

|

|

|
|

|

|

|

|

|
http://functions.wolfram.com/07.41.21.0003.01
|
|

|

|

|

|
|
|
|

|

|

|

|
|

|

|

|

|

|
Integrate[z^(\[Alpha] - 1) E^(z^2/4) ParabolicCylinderD[\[Nu], z], z] ==
((2^(\[Nu]/2) Sqrt[Pi] z^\[Alpha])/(\[Alpha] Gamma[(1 - \[Nu])/2]))
HypergeometricPFQ[{\[Alpha]/2, -(\[Nu]/2)}, {1/2, \[Alpha]/2 + 1},
z^2/2] - ((2^((\[Nu] + 1)/2) Sqrt[Pi] z^(\[Alpha] + 1))/
((\[Alpha] + 1) Gamma[-(\[Nu]/2)])) HypergeometricPFQ[
{(\[Alpha] + 1)/2, (1 - \[Nu])/2}, {3/2, (\[Alpha] + 3)/2}, z^2/2]
|
|

|

|

|

|
|

|

|

|

|
|

|

|

|

|

|
Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], "4"]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "/", "2"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Alpha]"]]], RowBox[List["\[Alpha]", " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["\[Alpha]", "2"], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["\[Alpha]", "2"], "+", "1"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["\[Alpha]", "+", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "+", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "2"]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]]]]]]]]
|
|

|

|

|

|
|

|

|

|

|
|

|

|

|

|

|
|

|

|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> D </mi> <annotation encoding='Mathematica'> TagBox["D", ParabolicCylinderD] </annotation> </semantics> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo>  </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> ν </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> α </mi> </msup> </mrow> <mrow> <mi> α </mi> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> α </mi> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mi> α </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["\[Alpha]", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> α </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "2"], HypergeometricPFQ, 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> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> ParabolicCylinderD </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <ci> ν </ci> <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> <power /> <ci> z </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <ci> α </ci> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </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 /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </list> <list> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <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> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <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> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <apply> <plus /> <ci> α </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|

|

|

|

|

| 
| 
| 
| 
| | 
| 
| 
| 
| 
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z_", "2"], "4"]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "/", "2"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Alpha]"]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["\[Alpha]", "2"], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["\[Alpha]", "2"], "+", "1"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], RowBox[List["\[Alpha]", " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["\[Alpha]", "+", "1"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "2"]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "+", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]]]]]]]]]]] |
| 
| 
| 
| 
|
|

|

|

|

|
Date Added to functions.wolfram.com (modification date)
|
|

|

|

|

|

|
|

|

|

|

|
|
 |
|
|