|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Hypergeometric Functions
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z]
Specific values
Specialized values
Cases with m==1
Case {m,n,p,q}={1,2,2,2}
|
|
|
|
|
|
|
http://functions.wolfram.com/07.34.03.0448.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
MeijerG[{{a, 3 a - 2 b - 3/2}, {}}, {{b}, {4 a - 3 b - 3}}, z] ==
((2^(5 - 6 a + 6 b) Gamma[1 - a + b] Gamma[5/2 - 3 a + 3 b])/
Gamma[4 - 4 a + 4 b]) z^b (4 + z)^(-(5/2) + 3 a - 3 b)
Hypergeometric2F1[5/6 - a + b, 7/6 - a + b, 5/2 - 2 a + 2 b,
(27 z^2)/(4 + z)^3]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["a", ",", RowBox[List[RowBox[List["3", " ", "a"]], "-", RowBox[List["2", " ", "b"]], "-", FractionBox["3", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "b", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["4", " ", "a"]], "-", RowBox[List["3", " ", "b"]], "-", "3"]], "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["5", "-", RowBox[List["6", " ", "a"]], "+", RowBox[List["6", " ", "b"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "a", "+", "b"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["5", "2"], "-", RowBox[List["3", " ", "a"]], "+", RowBox[List["3", " ", "b"]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["4", "-", RowBox[List["4", " ", "a"]], "+", RowBox[List["4", " ", "b"]]]], "]"]]], SuperscriptBox["z", "b"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["5", "2"]]], "+", RowBox[List["3", " ", "a"]], "-", RowBox[List["3", " ", "b"]]]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["5", "6"], "-", "a", "+", "b"]], ",", RowBox[List[FractionBox["7", "6"], "-", "a", "+", "b"]], ",", RowBox[List[FractionBox["5", "2"], "-", RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", "b"]]]], ",", FractionBox[RowBox[List["27", " ", SuperscriptBox["z", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", "z"]], ")"]], "3"]]]], "]"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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> 1 </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> 3 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> b </mi> <mo> , </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 3 </mn> </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["1", ",", "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["3", " ", "a"]], "-", RowBox[List["2", " ", "b"]], "-", FractionBox["3", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["4", " ", "a"]], "-", RowBox[List["3", " ", "b"]], "-", "3"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mfrac> <mn> 5 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> b </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mfrac> <mn> 5 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mfrac> <mn> 5 </mn> <mn> 6 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mfrac> <mn> 7 </mn> <mn> 6 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mfrac> <mn> 5 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mrow> <mn> 27 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["b", "-", "a", "+", FractionBox["5", "6"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "a", "+", FractionBox["7", "6"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["2", " ", "b"]], "-", RowBox[List["2", "a"]], "+", FractionBox["5", "2"]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["27", " ", SuperscriptBox["z", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "4"]], ")"]], "3"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> </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'> 3 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </list> <list /> </list> <list> <list> <ci> b </ci> </list> <list> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -3 </cn> </apply> </list> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> </apply> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> b </ci> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> b </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='rational'> 5 <sep /> 6 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='rational'> 7 <sep /> 6 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> </apply> <cn type='rational'> 5 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| 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["3", " ", "a_"]], "-", RowBox[List["2", " ", "b_"]], "-", FractionBox["3", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "b_", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["4", " ", "a_"]], "-", RowBox[List["3", " ", "b_"]], "-", "3"]], "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["5", "-", RowBox[List["6", " ", "a"]], "+", RowBox[List["6", " ", "b"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "a", "+", "b"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["5", "2"], "-", RowBox[List["3", " ", "a"]], "+", RowBox[List["3", " ", "b"]]]], "]"]]]], ")"]], " ", SuperscriptBox["z", "b"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["5", "2"]]], "+", RowBox[List["3", " ", "a"]], "-", RowBox[List["3", " ", "b"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["5", "6"], "-", "a", "+", "b"]], ",", RowBox[List[FractionBox["7", "6"], "-", "a", "+", "b"]], ",", RowBox[List[FractionBox["5", "2"], "-", RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", "b"]]]], ",", FractionBox[RowBox[List["27", " ", SuperscriptBox["z", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", "z"]], ")"]], "3"]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["4", "-", RowBox[List["4", " ", "a"]], "+", RowBox[List["4", " ", "b"]]]], "]"]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] | |
|
|
|