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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==2  Case {m,n,p,q}={2,1,2,2}   |  |  
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 | | http://functions.wolfram.com/07.34.03.0797.01 | 
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 | | MeijerG[{{a}, {a}}, {{-(3/4) + a, -(1/4) + a}, {}}, z] == 
  (4/Sqrt[Sqrt[z] + Sqrt[1 + z]]) z^(-(3/4) + a) 
   EllipticK[(Sqrt[z] - Sqrt[1 + z])/(Sqrt[z] + Sqrt[1 + z])] /; 
  !IntervalMemberQ[Interval[{-1, 0}], z] | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "a", "}"]], ",", RowBox[List["{", "a", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], "+", "a"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "+", "a"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["4", " "]], SqrtBox[RowBox[List[SqrtBox["z"], "+", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], "+", "a"]]], " ", RowBox[List["EllipticK", "[", FractionBox[RowBox[List[SqrtBox["z"], "-", SqrtBox[RowBox[List["1", "+", "z"]]]]], RowBox[List[SqrtBox["z"], "+", SqrtBox[RowBox[List["1", "+", "z"]]]]]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "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> 2 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mn> 1 </mn>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> ❘ </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> a </mi>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mfrac>  <mn> 3 </mn>  <mn> 4 </mn>  </mfrac>  </mrow>  <mo> , </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mn> 4 </mn>  </mfrac>  </mrow>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["a", MeijerG, Rule[Editable, True]], ",", TagBox["a", MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List["a", "-", FractionBox["3", "4"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", FractionBox["1", "4"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <mrow>  <mn> 4 </mn>  <mtext>   </mtext>  </mrow>  <msqrt>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> + </mo>  <msqrt>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  </mrow>  </msqrt>  </mfrac>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mfrac>  <mn> 3 </mn>  <mn> 4 </mn>  </mfrac>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> K </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mfrac>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> - </mo>  <msqrt>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  </mrow>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> + </mo>  <msqrt>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  </mrow>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mi> z </mi>  <mo> ∉ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </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> 2 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mn> 1 </mn>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> ❘ </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> a </mi>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mfrac>  <mn> 3 </mn>  <mn> 4 </mn>  </mfrac>  </mrow>  <mo> , </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mn> 4 </mn>  </mfrac>  </mrow>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["a", MeijerG, Rule[Editable, True]], ",", TagBox["a", MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List["a", "-", FractionBox["3", "4"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", FractionBox["1", "4"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <mrow>  <mn> 4 </mn>  <mtext>   </mtext>  </mrow>  <msqrt>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> + </mo>  <msqrt>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  </mrow>  </msqrt>  </mfrac>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mfrac>  <mn> 3 </mn>  <mn> 4 </mn>  </mfrac>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> K </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mfrac>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> - </mo>  <msqrt>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  </mrow>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> + </mo>  <msqrt>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  </mrow>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mi> z </mi>  <mo> ∉ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </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["{", "a_", "}"]], ",", RowBox[List["{", "a_", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], "+", "a_"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "+", "a_"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["4", " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], "+", "a"]]], " ", RowBox[List["EllipticK", "[", FractionBox[RowBox[List[SqrtBox["z"], "-", SqrtBox[RowBox[List["1", "+", "z"]]]]], RowBox[List[SqrtBox["z"], "+", SqrtBox[RowBox[List["1", "+", "z"]]]]]], "]"]]]], SqrtBox[RowBox[List[SqrtBox["z"], "+", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "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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