Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
MeijerG




Mathematica Notation

Traditional Notation









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}





http://functions.wolfram.com/07.34.03.0777.01









  


  








Input Form





MeijerG[{{a}, {1/2 + a}}, {{b, -1 + 4 a - 3 b}, {}}, z] == ((2 3^(-1 + 3 a - 3 b) Gamma[3 a - 3 b] Gamma[1 - a + b])/Sqrt[Pi]) z^(-1 + 4 a - 3 b) (4 + 3 z)^(-1 - 3 a + 3 b) (8 + 9 z) Hypergeometric2F1[1/3 + a - b, 2/3 + a - b, 3/2, (8 + 9 z)^2/(4 + 3 z)^3] /; Abs[z] > 1 || Re[z] >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "a", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], "+", "a"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["b", ",", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", "a"]], "-", RowBox[List["3", " ", "b"]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["3", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["3", " ", "a"]], "-", RowBox[List["3", " ", "b"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["3", " ", "a"]], "-", RowBox[List["3", " ", "b"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "a", "+", "b"]], "]"]], " "]], SqrtBox["\[Pi]"]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", "a"]], "-", RowBox[List["3", " ", "b"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List["3", " ", "z"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["3", " ", "a"]], "+", RowBox[List["3", " ", "b"]]]]], " ", RowBox[List["(", RowBox[List["8", "+", RowBox[List["9", " ", "z"]]]], ")"]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "3"], "+", "a", "-", "b"]], ",", RowBox[List[FractionBox["2", "3"], "+", "a", "-", "b"]], ",", FractionBox["3", "2"], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["8", "+", RowBox[List["9", " ", "z"]]]], ")"]], "2"], SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List["3", " ", "z"]]]], ")"]], "3"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]], "\[Or]", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[GreaterEqual]", "0"]]]]]]]]








MathML Form







<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> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mi> a </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> b </mi> <mo> , </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;2&quot;]], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;1&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;z&quot;, MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[&quot;a&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[&quot;b&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[RowBox[List[&quot;4&quot;, &quot; &quot;, &quot;a&quot;]], &quot;-&quot;, RowBox[List[&quot;3&quot;, &quot; &quot;, &quot;b&quot;]], &quot;-&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <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[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;b&quot;, &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;3&quot;]]], Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;b&quot;, &quot;+&quot;, FractionBox[&quot;2&quot;, &quot;3&quot;]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[FractionBox[&quot;3&quot;, &quot;2&quot;], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[SuperscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;9&quot;, &quot; &quot;, &quot;z&quot;]], &quot;+&quot;, &quot;8&quot;]], &quot;)&quot;]], &quot;2&quot;], SuperscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;3&quot;, &quot; &quot;, &quot;z&quot;]], &quot;+&quot;, &quot;4&quot;]], &quot;)&quot;]], &quot;3&quot;]], Hypergeometric2F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8805; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> MeijerG </ci> <list> <list> <ci> a </ci> </list> <list> <apply> <plus /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </list> </list> <list> <list> <ci> b </ci> <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'> -1 </cn> </apply> </list> <list /> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <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> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </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'> 3 </cn> <ci> b </ci> </apply> </apply> </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> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <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'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -3 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> z </ci> </apply> <cn type='integer'> 8 </cn> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> z </ci> </apply> <cn type='integer'> 8 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <gt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <geq /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "a_", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], "+", "a_"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["b_", ",", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", "a_"]], "-", RowBox[List["3", " ", "b_"]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["3", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["3", " ", "a"]], "-", RowBox[List["3", " ", "b"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["3", " ", "a"]], "-", RowBox[List["3", " ", "b"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "a", "+", "b"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", "a"]], "-", RowBox[List["3", " ", "b"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List["3", " ", "z"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["3", " ", "a"]], "+", RowBox[List["3", " ", "b"]]]]], " ", RowBox[List["(", RowBox[List["8", "+", RowBox[List["9", " ", "z"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "3"], "+", "a", "-", "b"]], ",", RowBox[List[FractionBox["2", "3"], "+", "a", "-", "b"]], ",", FractionBox["3", "2"], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["8", "+", RowBox[List["9", " ", "z"]]]], ")"]], "2"], SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List["3", " ", "z"]]]], ")"]], "3"]]]], "]"]]]], SqrtBox["\[Pi]"]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]], "||", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[GreaterEqual]", "0"]]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29