Parabolic cylinder function: Differentiation (formula 07.41.20.0028)

ParabolicCylinderD

$Mathematica Notation$

Hypergeometric Functions

ParabolicCylinderD[nu,z]

Differentiation

Fractional integro-differentiation

With respect to z

http://functions.wolfram.com/07.41.20.0028.01

Input Form

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

Standard Form

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MathML Form

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1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> α </mi> <mn> 2 </mn> </mfrac> </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[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", 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Rule Form

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

2007-05-02