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AppellF1






Mathematica Notation

Traditional Notation









Hypergeometric Functions > AppellF1[a,b1,b2,c,z1,z2] > Representations through more general functions > Through Meijer G > Classical cases for the direct function itself





http://functions.wolfram.com/07.36.26.0003.01









  


  










Input Form





AppellF1[a, Subscript[b, 1], Subscript[b, 2], c, Subscript[z, 1], Subscript[z, 2]] == (Gamma[c]/(Gamma[a] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])) MeijerG[{{{1 - a}, {}}, {{}, {1 - c}}}, {{{1 - Subscript[b, 1]}, {}}, {{0}, {}}}, {{{1 - Subscript[b, 2]}, {}}, {{0}, {}}}, -Subscript[z, 1], -Subscript[z, 2]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "c", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Gamma", "[", "c", "]"]], RowBox[List[" ", RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["b", "2"], "]"]]]]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", "a"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "c"]], "}"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["b", "1"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", "}"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["b", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", "}"]]]], "}"]]]], "}"]], ",", RowBox[List["-", SubscriptBox["z", "1"]]], ",", RowBox[List["-", SubscriptBox["z", "2"]]]]], "]"]]]]]]]]










MathML Form







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</mo> <mtable> <mtr> <mtd> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> </mtr> </mtable> <mo> &#10072; </mo> <mrow> <mo> - </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> InterpretationBox </ci> <apply> <ci> SubscriptBox </ci> <ms> F </ms> <ms> 1 </ms> </apply> <ci> AppellF1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> <apply> <ci> Rule </ci> <ci> Selectable </ci> <false /> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> ; </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <ms> c </ms> <ms> ; </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> c </ms> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> a </ms> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> G </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> : </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> : </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 0 </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> : </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> : </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> [ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> a </ms> </list> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> c </ms> </list> </apply> </list> </list> </apply> <ms> &#10072; </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> </list> </apply> </list> <list> <ms> 0 </ms> </list> </list> </apply> <ms> &#10072; </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 2 </ms> </apply> </list> </apply> </list> <list> <ms> 0 </ms> </list> </list> </apply> <ms> &#10072; </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 1 </ms> </apply> </list> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </list> </apply> </list> </apply> <ms> ] </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AppellF1", "[", RowBox[List["a_", ",", SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"], ",", "c_", ",", SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", "a"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "c"]], "}"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["bb", "1"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", "}"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["bb", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", "}"]]]], "}"]]]], "}"]], ",", RowBox[List["-", SubscriptBox["zz", "1"]]], ",", RowBox[List["-", SubscriptBox["zz", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "2"], "]"]]]]]]]]]










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





2001-10-29





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