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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}={2,3,4,2}   |  |  
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 | | http://functions.wolfram.com/07.34.03.0940.01 | 
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 | | MeijerG[{{a, c, 2 b - c}, {-a + 2 b}}, {{b, -(1/2) + b}, {}}, z] == 
  Pi^(5/2) z^(-1 + b) (BesselJ[-a + c, 1/Sqrt[z]] BesselY[-a + 2 b - c, 
      1/Sqrt[z]] - BesselJ[-a + 2 b - c, 1/Sqrt[z]] 
     BesselY[-a + c, 1/Sqrt[z]]) Csc[2 (-b + c) 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", ",", "c", ",", RowBox[List[RowBox[List["2", " ", "b"]], "-", "c"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["2", " ", "b"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["b", ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "b"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["5", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "+", "c"]], ",", FractionBox["1", SqrtBox["z"]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["2", " ", "b"]], "-", "c"]], ",", FractionBox["1", SqrtBox["z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["2", " ", "b"]], "-", "c"]], ",", FractionBox["1", SqrtBox["z"]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "+", "c"]], ",", FractionBox["1", SqrtBox["z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", "\[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> 4 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mn> 3 </mn>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> ❘ </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> c </mi>  <mo> , </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> - </mo>  <mi> c </mi>  </mrow>  <mo> , </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> - </mo>  <mi> a </mi>  </mrow>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mi> b </mi>  <mo> , </mo>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "2"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["a", MeijerG, Rule[Editable, True]], ",", TagBox["c", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "b"]], "-", "c"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "b"]], "-", "a"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", FractionBox["1", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <msup>  <mi> π </mi>  <mrow>  <mn> 5 </mn>  <mo> / </mo>  <mn> 2 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> csc </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mi> π </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <msub>  <mi> J </mi>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> a </mi>  </mrow>  </msub>  <mo> ( </mo>  <mfrac>  <mn> 1 </mn>  <msqrt>  <mi> z </mi>  </msqrt>  </mfrac>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msub>  <mi> Y </mi>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> - </mo>  <mi> a </mi>  <mo> - </mo>  <mi> c </mi>  </mrow>  </msub>  <mo> ( </mo>  <mfrac>  <mn> 1 </mn>  <msqrt>  <mi> z </mi>  </msqrt>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  <mo> - </mo>  <mrow>  <mrow>  <msub>  <mi> J </mi>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> - </mo>  <mi> a </mi>  <mo> - </mo>  <mi> c </mi>  </mrow>  </msub>  <mo> ( </mo>  <mfrac>  <mn> 1 </mn>  <msqrt>  <mi> z </mi>  </msqrt>  </mfrac>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msub>  <mi> Y </mi>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> a </mi>  </mrow>  </msub>  <mo> ( </mo>  <mfrac>  <mn> 1 </mn>  <msqrt>  <mi> z </mi>  </msqrt>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mtext>   </mtext>  <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> 4 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mn> 3 </mn>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> ❘ </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> c </mi>  <mo> , </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> - </mo>  <mi> c </mi>  </mrow>  <mo> , </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> - </mo>  <mi> a </mi>  </mrow>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mi> b </mi>  <mo> , </mo>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "2"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["a", MeijerG, Rule[Editable, True]], ",", TagBox["c", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "b"]], "-", "c"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "b"]], "-", "a"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", FractionBox["1", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <msup>  <mi> π </mi>  <mrow>  <mn> 5 </mn>  <mo> / </mo>  <mn> 2 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> csc </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mi> π </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <msub>  <mi> J </mi>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> a </mi>  </mrow>  </msub>  <mo> ( </mo>  <mfrac>  <mn> 1 </mn>  <msqrt>  <mi> z </mi>  </msqrt>  </mfrac>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msub>  <mi> Y </mi>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> - </mo>  <mi> a </mi>  <mo> - </mo>  <mi> c </mi>  </mrow>  </msub>  <mo> ( </mo>  <mfrac>  <mn> 1 </mn>  <msqrt>  <mi> z </mi>  </msqrt>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  <mo> - </mo>  <mrow>  <mrow>  <msub>  <mi> J </mi>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> - </mo>  <mi> a </mi>  <mo> - </mo>  <mi> c </mi>  </mrow>  </msub>  <mo> ( </mo>  <mfrac>  <mn> 1 </mn>  <msqrt>  <mi> z </mi>  </msqrt>  </mfrac>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msub>  <mi> Y </mi>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> a </mi>  </mrow>  </msub>  <mo> ( </mo>  <mfrac>  <mn> 1 </mn>  <msqrt>  <mi> z </mi>  </msqrt>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mtext>   </mtext>  <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_", ",", "c_", ",", RowBox[List[RowBox[List["2", " ", "b_"]], "-", "c_"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", "a_"]], "+", RowBox[List["2", " ", "b_"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["b_", ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "b_"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["5", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "+", "c"]], ",", FractionBox["1", SqrtBox["z"]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["2", " ", "b"]], "-", "c"]], ",", FractionBox["1", SqrtBox["z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["2", " ", "b"]], "-", "c"]], ",", FractionBox["1", SqrtBox["z"]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "+", "c"]], ",", FractionBox["1", SqrtBox["z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", "\[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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