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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}={0,2,2,2}   |  |  
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 | | http://functions.wolfram.com/07.34.03.0116.01 | 
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 | | MeijerG[{{a, 2 b - a + 1}, {}}, {{}, {b, b + 1/2}}, z] == 
 (2^(2 (b - a))/(Sqrt[Pi] Sqrt[z - 1])) z^b 
  ((Sqrt[Sqrt[z] + 1] - Sqrt[Sqrt[z] - 1])^(4 (a - b) - 2) + 
   (Sqrt[Sqrt[z] + 1] + Sqrt[Sqrt[z] - 1])^(4 (a - b) - 2)) 
  UnitStep[Abs[z] - 1] | 
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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>  <semantics>  <mrow>  <msubsup>  <mi> G </mi>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mn> 2 </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>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> - </mo>  <mi> a </mi>  <mo> + </mo>  <mn> 1 </mn>  </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["2", ",", "2"]], 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[RowBox[List["2", " ", "b"]], "-", "a", "+", "1"]], 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>  <mfrac>  <mrow>  <msup>  <mn> 2 </mn>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mi> a </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </msup>  <mo> ⁢ </mo>  <mtext>    </mtext>  <mrow>  <semantics>  <mi> θ </mi>  <annotation-xml encoding='MathML-Content'>  <ci> UnitStep </ci>  </annotation-xml>  </semantics>  <mo> ( </mo>  <mrow>  <mrow>  <semantics>  <mo> ❘ </mo>  <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation>  </semantics>  <mi> z </mi>  <semantics>  <mo> ❘ </mo>  <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation>  </semantics>  </mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mrow>  <msqrt>  <mi> π </mi>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mi> z </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mi> b </mi>  </msup>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <msqrt>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  <mo> - </mo>  <msqrt>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mrow>  <mn> 4 </mn>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> - </mo>  <mn> 2 </mn>  </mrow>  </msup>  <mo> + </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <msqrt>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  <mo> + </mo>  <msqrt>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mrow>  <mn> 4 </mn>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> - </mo>  <mn> 2 </mn>  </mrow>  </msup>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <eq />  <apply>  <ci> MeijerG </ci>  <list>  <list>  <ci> a </ci>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> b </ci>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> a </ci>  </apply>  <cn type='integer'> 1 </cn>  </apply>  </list>  <list />  </list>  <list>  <list />  <list>  <ci> b </ci>  <apply>  <plus />  <ci> b </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </list>  </list>  <ci> z </ci>  </apply>  <apply>  <times />  <apply>  <times />  <apply>  <power />  <cn type='integer'> 2 </cn>  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <plus />  <ci> b </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> a </ci>  </apply>  </apply>  </apply>  </apply>  <apply>  <ci> UnitStep </ci>  <apply>  <plus />  <apply>  <abs />  <ci> z </ci>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <power />  <apply>  <times />  <apply>  <power />  <pi />  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <ci> z </ci>  <cn type='integer'> -1 </cn>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <power />  <ci> z </ci>  <ci> b </ci>  </apply>  <apply>  <plus />  <apply>  <power />  <apply>  <plus />  <apply>  <power />  <apply>  <plus />  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <power />  <apply>  <plus />  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  </apply>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 4 </cn>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  </apply>  <cn type='integer'> -2 </cn>  </apply>  </apply>  <apply>  <power />  <apply>  <plus />  <apply>  <power />  <apply>  <plus />  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 4 </cn>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  </apply>  <cn type='integer'> -2 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  </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[RowBox[List["2", " ", "b_"]], "-", "a_", "+", "1"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["b_", ",", RowBox[List["b_", "+", FractionBox["1", "2"]]]]], "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "-", "a"]], ")"]]]]], " ", SuperscriptBox["z", "b"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[SqrtBox["z"], "+", "1"]]], "-", SqrtBox[RowBox[List[SqrtBox["z"], "-", "1"]]]]], ")"]], RowBox[List[RowBox[List["4", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]]]], "-", "2"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[SqrtBox["z"], "+", "1"]]], "+", SqrtBox[RowBox[List[SqrtBox["z"], "-", "1"]]]]], ")"]], RowBox[List[RowBox[List["4", " ", RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]]]], "-", "2"]]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "-", "1"]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", SqrtBox[RowBox[List["z", "-", "1"]]]]]]]]]] | 
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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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