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| Hypergeometric Functions  MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z]  Specific values  Specialized values  Cases with m==0  Case {m,n,p,q}={0,2,3,0}   |  |  
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 | | http://functions.wolfram.com/07.34.03.0169.01 | 
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 | | MeijerG[{{a, a - 1/3}, {a + 1/3}}, {{}, {}}, z] == 
  (((2 z^(-(2/3) + a))/Sqrt[3]) Sin[(3 Sqrt[3])/(2 z^(1/3))])/
   E^(3/(2 z^(1/3))) /;  !IntervalMemberQ[Interval[{-Infinity, 0}], z] | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["a", ",", RowBox[List["a", "-", FractionBox["1", "3"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["a", "+", FractionBox["1", "3"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["2", SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["2", "3"]]], "+", "a"]]]]], SqrtBox["3"]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["3", RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]]]]]], RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", SqrtBox["3"]]], RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]] | 
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   <math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'>  <semantics>  <mrow>  <mrow>  <semantics>  <mrow>  <msubsup>  <mi> G </mi>  <mrow>  <mn> 3 </mn>  <mo> , </mo>  <mn> 0 </mn>  </mrow>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> ❘ </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mn> 3 </mn>  </mfrac>  </mrow>  <mo> , </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mfrac>  <mn> 1 </mn>  <mn> 3 </mn>  </mfrac>  </mrow>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "0"]], RowBox[List["0", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["a", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", FractionBox["1", "3"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", FractionBox["1", "3"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mfrac>  <mn> 2 </mn>  <mn> 3 </mn>  </mfrac>  </mrow>  </msup>  </mrow>  <msqrt>  <mn> 3 </mn>  </msqrt>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mi> exp </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mo> - </mo>  <mfrac>  <mn> 3 </mn>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mroot>  <mi> z </mi>  <mn> 3 </mn>  </mroot>  </mrow>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> sin </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mfrac>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <msqrt>  <mn> 3 </mn>  </msqrt>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mroot>  <mi> z </mi>  <mn> 3 </mn>  </mroot>  </mrow>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mi> z </mi>  <mo> ∉ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> - </mo>  <mi> ∞ </mi>  </mrow>  <mo> , </mo>  <mn> 0 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <mrow>  <mrow>  <semantics>  <mrow>  <msubsup>  <mi> G </mi>  <mrow>  <mn> 3 </mn>  <mo> , </mo>  <mn> 0 </mn>  </mrow>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> ❘ </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mn> 3 </mn>  </mfrac>  </mrow>  <mo> , </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mfrac>  <mn> 1 </mn>  <mn> 3 </mn>  </mfrac>  </mrow>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "0"]], RowBox[List["0", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["a", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", FractionBox["1", "3"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", FractionBox["1", "3"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mfrac>  <mn> 2 </mn>  <mn> 3 </mn>  </mfrac>  </mrow>  </msup>  </mrow>  <msqrt>  <mn> 3 </mn>  </msqrt>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mi> exp </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mo> - </mo>  <mfrac>  <mn> 3 </mn>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mroot>  <mi> z </mi>  <mn> 3 </mn>  </mroot>  </mrow>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> sin </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mfrac>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <msqrt>  <mn> 3 </mn>  </msqrt>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mroot>  <mi> z </mi>  <mn> 3 </mn>  </mroot>  </mrow>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mi> z </mi>  <mo> ∉ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> - </mo>  <mi> ∞ </mi>  </mrow>  <mo> , </mo>  <mn> 0 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </annotation-xml>  </semantics>  </math> 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", RowBox[List["a_", "-", FractionBox["1", "3"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["a_", "+", FractionBox["1", "3"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["2", "3"]]], "+", "a"]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["3", RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]]]]]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", SqrtBox["3"]]], RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]]], "]"]]]], SqrtBox["3"]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]] | 
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 | | Date Added to functions.wolfram.com (modification date) | 
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 | | MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] |  |  | 
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