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ParabolicCylinderD




Mathematica Notation

Traditional Notation









Hypergeometric Functions > ParabolicCylinderD[nu,z] > Representations through more general functions > Through hypergeometric functions > Involving 1F~1





http://functions.wolfram.com/07.41.26.0004.01









  


  








Input Form





ParabolicCylinderD[\[Nu], z] == (2^(\[Nu]/2) Pi ((1/Gamma[(1 - \[Nu])/2]) Hypergeometric1F1Regularized[ -(\[Nu]/2), 1/2, z^2/2] - (z/(Sqrt[2] Gamma[-(\[Nu]/2)])) Hypergeometric1F1Regularized[(1 - \[Nu])/2, 3/2, z^2/2]))/E^(z^2/4)








Standard Form





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MathML Form







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</mo> <mn> 1 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <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[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;1&quot;], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], &quot;1&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;\[Nu]&quot;, &quot;2&quot;]]], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[SuperscriptBox[&quot;z&quot;, &quot;2&quot;], &quot;2&quot;], Hypergeometric1F1Regularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1Regularized] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mtext> </mtext> </mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "/", "2"]]], " ", "\[Pi]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Hypergeometric1F1Regularized", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]], RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]]], "-", FractionBox[RowBox[List["z", " ", RowBox[List["Hypergeometric1F1Regularized", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", FractionBox["3", "2"], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], RowBox[List[SqrtBox["2"], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]]]]]]], ")"]]]]]]]]


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





2007-05-02