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ParabolicCylinderD






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ParabolicCylinderD[nu,z] > Integral representations > Contour integral representations





http://functions.wolfram.com/07.41.07.0019.01









  


  










Input Form





ParabolicCylinderD[\[Nu], z] == ((2^((\[Nu] + 1)/2) Gamma[\[Nu]/2 + 1])/ (2 Pi I)) ContourIntegrate[(1 + t)^(-(\[Nu]/2) - 1) (1 - t)^((\[Nu] - 1)/2) E^((z^2 t)/4), {t, L}] /; Arg[z] < Pi/4 && Arg[1 + t] <= Pi










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], "]"]]]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List["ContourIntegrate", "[", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "t"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], FractionBox[RowBox[List["\[Nu]", "-", "1"]], "2"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["z", "2"], " ", "t"]], "4"]]]], ",", RowBox[List["{", RowBox[List["t", ",", "L"]], "}"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Arg", "[", "z", "]"]], "<", FractionBox["\[Pi]", "4"]]], "&&", RowBox[List[RowBox[List["Arg", "[", RowBox[List["1", "+", "t"]], "]"]], "\[LessEqual]", "\[Pi]"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> D </mi> <annotation encoding='Mathematica'> TagBox[&quot;D&quot;, ParabolicCylinderD] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mfrac> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mo> &#8747; </mo> <mi> L </mi> </msub> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mn> 4 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> arg </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &lt; </mo> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mi> &#960; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> D </ms> <ci> ParabolicCylinderD </ci> </apply> <ms> &#957; </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#63449; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> 2 </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#957; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> 2 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> &#957; </ms> <ms> 2 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#960; </ms> <ms> &#8520; </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8747; </ms> </apply> <ms> L </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> t </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> &#957; </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> t </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#957; </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> 2 </ms> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> &#8519; </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> t </ms> </list> </apply> <ms> 4 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8518; </ms> <ms> t </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> arg </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> &lt; </ms> <apply> <ci> FractionBox </ci> <ms> &#960; </ms> <ms> 4 </ms> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> arg </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> t </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> &#8804; </ms> <ms> &#960; </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], "]"]]]], ")"]], " ", RowBox[List["ContourIntegrate", "[", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "t"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], FractionBox[RowBox[List["\[Nu]", "-", "1"]], "2"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["z", "2"], " ", "t"]], "4"]]]], ",", RowBox[List["{", RowBox[List["t", ",", "L"]], "}"]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Arg", "[", "z", "]"]], "<", FractionBox["\[Pi]", "4"]]], "&&", RowBox[List[RowBox[List["Arg", "[", RowBox[List["1", "+", "t"]], "]"]], "\[LessEqual]", "\[Pi]"]]]]]]]]]]










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





2007-05-02





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