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Exp






Mathematica Notation

Traditional Notation









Elementary Functions > Exp[z] > Representations through more general functions > Through Meijer G > Generalized cases involving parabolic cylinder function D





http://functions.wolfram.com/01.03.26.0212.01









  


  










Input Form





E^z^2 ParabolicCylinderD[\[Nu], 2 z] == (2^(-1 - \[Nu]/2)/(Sqrt[Pi] Gamma[-\[Nu]])) MeijerG[{{1 + \[Nu]/2}, {}}, {{0, 1/2}, {}}, Sqrt[2] z, 1/2]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", SuperscriptBox["z", "2"]], RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", RowBox[List["2", "z"]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["\[Nu]", "2"]]]], RowBox[List[SqrtBox["\[Pi]"], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List[SqrtBox["2"], "z"]], ",", FractionBox["1", "2"]]], "]"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> &#8519; </mi> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msup> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> D </mi> <annotation encoding='Mathematica'> TagBox[&quot;D&quot;, ParabolicCylinderD] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> G </mi> <annotation encoding='Mathematica'> TagBox[&quot;G&quot;, MeijerG] </annotation> </semantics> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[TagBox[RowBox[List[SqrtBox[&quot;2&quot;], &quot; &quot;, &quot;z&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]]]], MeijerG] </annotation> </semantics> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <semantics> <mrow> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[FractionBox[&quot;\[Nu]&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mtd> </mtr> <mtr> <mtd> <mrow> <semantics> <mn> 0 </mn> <annotation encoding='Mathematica'> TagBox[&quot;0&quot;, MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> &#8519; 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</ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SqrtBox </ci> <ms> 2 </ms> </apply> <ms> </ms> <ms> z </ms> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ci> MeijerG </ci> </apply> <ms> &#10072; </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> &#957; </ms> <ms> 2 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> 0 </ms> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> </list> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </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[SuperscriptBox["\[ExponentialE]", SuperscriptBox["z_", "2"]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]_", ",", RowBox[List["2", " ", "z_"]]]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["\[Nu]", "2"]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List[SqrtBox["2"], " ", "z"]], ",", FractionBox["1", "2"]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]]]]]]










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





2007-05-02