Parabolic cylinder function: Representations through more general functions (formula 07.41.26.0003)

ParabolicCylinderD

$Mathematica Notation$

Hypergeometric Functions

ParabolicCylinderD[nu,z]

Representations through more general functions

Through hypergeometric functions

Involving _pF_q

http://functions.wolfram.com/07.41.26.0003.01

Input Form

ParabolicCylinderD[\[Nu], I z] ParabolicCylinderD[\[Nu], z] == ((2^\[Nu] Pi)/Gamma[(1 - \[Nu])/2]^2) HypergeometricPFQ[ {(1 + \[Nu])/2, -(\[Nu]/2)}, {1/4, 1/2, 3/4}, z^4/16] - (((-1)^(1/4) Sqrt[Pi] z)/Gamma[-\[Nu]]) HypergeometricPFQ[ {(1 - 2 \[Nu])/4, (3 + 2 \[Nu])/4}, {1/2, 3/4, 5/4}, z^4/16] + ((I 2^(1 + \[Nu]) Pi z^2)/Gamma[-(\[Nu]/2)]^2) HypergeometricPFQ[{(1 - \[Nu])/2, 1 + \[Nu]/2}, {3/4, 5/4, 3/2}, z^4/16] + (((-1)^(3/4) Sqrt[Pi] (1 + 2 \[Nu]) z^3)/(6 Gamma[-\[Nu]])) HypergeometricPFQ[{(3 - 2 \[Nu])/4, (5 + 2 \[Nu])/4}, {5/4, 3/2, 7/4}, z^4/16]

Standard Form

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MathML Form

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</apply> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <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> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <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> <plus /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list> <cn type='rational'> 3 <sep /> 4 </cn> <cn type='rational'> 5 <sep /> 4 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </list> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 16 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 5 </cn> </apply> </apply> </list> <list> <cn type='rational'> 5 <sep /> 4 </cn> <cn type='rational'> 3 <sep /> 2 </cn> <cn type='rational'> 7 <sep /> 4 </cn> </list> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 16 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02