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ParabolicCylinderD






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ParabolicCylinderD[nu,z] > Integral representations > On the real axis > Of the direct function





http://functions.wolfram.com/07.41.07.0006.01









  


  










Input Form





ParabolicCylinderD[\[Nu], z] == ((2^(-\[Nu] - 1)/(Gamma[-\[Nu]] z^\[Nu])) Integrate[(t^(-\[Nu] - 1) (Sqrt[2 t + 1] + 1)^(\[Nu] + 1))/ (2 t + 1)^(1/2)/E^(z^2 t), {t, 0, Infinity}])/E^(z^2/4) /; Re[\[Nu]] < 0 && Re[z^2] > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", SuperscriptBox["z", "\[Nu]"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["t", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "t"]], "+", "1"]]], "+", "1"]], ")"]], RowBox[List["\[Nu]", "+", "1"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "t"]], "+", "1"]], ")"]], RowBox[List["1", "/", "2"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], " ", "t"]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "<", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", SuperscriptBox["z", "2"], "]"]], ">", "0"]]]]]]]]










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> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> &#957; </mi> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mfrac> <mrow> <msup> <mi> t </mi> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> &#8290; </mo> <mi> t </mi> </mrow> </msup> </mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> ParabolicCylinderD </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <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> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> t </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> t </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> t </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <real /> <ci> &#957; </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </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[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["t", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "t"]], "+", "1"]]], "+", "1"]], ")"]], RowBox[List["\[Nu]", "+", "1"]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], " ", "t"]]]]], SqrtBox[RowBox[List[RowBox[List["2", " ", "t"]], "+", "1"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", SuperscriptBox["z", "\[Nu]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "<", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", SuperscriptBox["z", "2"], "]"]], ">", "0"]]]]]]]]]]










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





2007-05-02





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