|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Hypergeometric Functions
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z]
Specific values
Specialized values
Cases with m==3
Case {m,n,p,q}={3,1,2,4}
|
|
|
|
|
|
|
http://functions.wolfram.com/07.34.03.0989.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
MeijerG[{{a}, {c}}, {{b, (a + c - 1)/2, (a + c)/2}, {a - b + c - 1}}, z] ==
((2^(2 - a + 2 b - c) Sqrt[Pi] Gamma[1 - a + b])/Gamma[2 - a + 2 b - c]) z^b
Hypergeometric1F1[1 - a + b, 2 - a + 2 b - c, -2 Sqrt[z]]
HypergeometricU[1 - a + b, 2 - a + 2 b - c, 2 Sqrt[z]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "a", "}"]], ",", RowBox[List["{", "c", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["b", ",", FractionBox[RowBox[List["a", "+", "c", "-", "1"]], "2"], ",", FractionBox[RowBox[List["a", "+", "c"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["a", "-", "b", "+", "c", "-", "1"]], "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["2", "-", "a", "+", RowBox[List["2", " ", "b"]], "-", "c"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "a", "+", "b"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "a", "+", RowBox[List["2", " ", "b"]], "-", "c"]], "]"]]], " ", SuperscriptBox["z", "b"], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List["1", "-", "a", "+", "b"]], ",", RowBox[List["2", "-", "a", "+", RowBox[List["2", " ", "b"]], "-", "c"]], ",", RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List["1", "-", "a", "+", "b"]], ",", RowBox[List["2", "-", "a", "+", RowBox[List["2", " ", "b"]], "-", "c"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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> 4 </mn> </mrow> <mrow> <mn> 3 </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> c </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> b </mi> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> c </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "4"]], RowBox[List["3", ",", "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["c", MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "c", "-", "1"]], "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "c"]], "2"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", "b", "+", "c", "-", "1"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> b </mi> </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> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["b", "-", "a", "+", "1"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["2", " ", "b"]], "-", "a", "-", "c", "+", "2"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox["z"]]], Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox["U", HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> , </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> MeijerG </ci> <list> <list> <ci> a </ci> </list> <list> <ci> c </ci> </list> </list> <list> <list> <ci> b </ci> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> c </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> c </ci> <cn type='integer'> -1 </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'> 2 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <ci> b </ci> </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> <power /> <apply> <ci> Gamma </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> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric1F1 </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricU </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </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["{", "a_", "}"]], ",", RowBox[List["{", "c_", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["b_", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a_", "+", "c_", "-", "1"]], ")"]]]], ",", FractionBox[RowBox[List["a_", "+", "c_"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["a_", "-", "b_", "+", "c_", "-", "1"]], "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["2", "-", "a", "+", RowBox[List["2", " ", "b"]], "-", "c"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "a", "+", "b"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "b"], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List["1", "-", "a", "+", "b"]], ",", RowBox[List["2", "-", "a", "+", RowBox[List["2", " ", "b"]], "-", "c"]], ",", RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List["1", "-", "a", "+", "b"]], ",", RowBox[List["2", "-", "a", "+", RowBox[List["2", " ", "b"]], "-", "c"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "a", "+", RowBox[List["2", " ", "b"]], "-", "c"]], "]"]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] | |
|
|
|