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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}={0,2,3,1}   |  |  
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 | | http://functions.wolfram.com/07.34.03.0184.01 | 
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 | | MeijerG[{{a, a + 1/2}, {b}}, {{}, {b}}, z] == 
  (z^(-(1/2) + a) Cos[Pi (a - b) + 2/Sqrt[z]])/Sqrt[Pi] /; 
  !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", "2"]]]]], "}"]], ",", RowBox[List["{", "b", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "b", "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "a"]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["\[Pi]", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]]]], "+", FractionBox["2", SqrtBox["z"]]]], "]"]]]], SqrtBox["\[Pi]"]]]], "/;", 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> 1 </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> 2 </mn>  </mfrac>  </mrow>  <mo> , </mo>  <mi> b </mi>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mi> b </mi>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "1"]], 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", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox["b", MeijerG, Rule[Editable, True]]]]], List[TagBox["b", MeijerG, Rule[Editable, True]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <msup>  <mi> z </mi>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  </msup>  <msqrt>  <mi> π </mi>  </msqrt>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> + </mo>  <mfrac>  <mn> 2 </mn>  <msqrt>  <mi> z </mi>  </msqrt>  </mfrac>  </mrow>  <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> 1 </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> 2 </mn>  </mfrac>  </mrow>  <mo> , </mo>  <mi> b </mi>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mi> b </mi>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "1"]], 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", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox["b", MeijerG, Rule[Editable, True]]]]], List[TagBox["b", MeijerG, Rule[Editable, True]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <msup>  <mi> z </mi>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  </msup>  <msqrt>  <mi> π </mi>  </msqrt>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> + </mo>  <mfrac>  <mn> 2 </mn>  <msqrt>  <mi> z </mi>  </msqrt>  </mfrac>  </mrow>  <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", "2"]]]]], "}"]], ",", RowBox[List["{", "b_", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "b_", "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "a"]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]]]], "+", FractionBox["2", SqrtBox["z"]]]], "]"]]]], SqrtBox["\[Pi]"]], "/;", 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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