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AppellF1






Mathematica Notation

Traditional Notation









Hypergeometric Functions > AppellF1[a,b1,b2,c,z1,z2] > Integral representations > Multiple integral representations





http://functions.wolfram.com/07.36.07.0003.01









  


  










Input Form





AppellF1[a, Subscript[b, 1], Subscript[b, 2], c, Subscript[z, 1], Subscript[z, 2]] == (Gamma[c]/(Gamma[c - Subscript[b, 1] - Subscript[b, 2]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])) Integrate[(Subscript[z, 1]^(Subscript[b, 1] - 1) Subscript[z, 2]^(Subscript[b, 2] - 1) (1 - x - y)^ (c - Subscript[b, 1] - Subscript[b, 2] - 1))/ (1 - x Subscript[z, 1] - y Subscript[z, 2])^a, {x, 0, 1}, {y, 0, 1 - x}] /; Re[Subscript[b, 1]] > 0 && Re[Subscript[b, 2]] > 0 && Re[c - Subscript[b, 1] - Subscript[b, 2]] > 0










Standard Form





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MathML Form







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</mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mn> 1 </mn> </msubsup> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> x </mi> </mrow> </msubsup> <mrow> <msubsup> <mi> z </mi> <mn> 1 </mn> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> x </mi> </mrow> <mo> - </mo> <mi> y </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> c </mi> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; 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</mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> AppellF1 </ci> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <ci> c </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> y </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </uplimit> <apply> <int /> <bvar> <ci> x </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> y </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </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]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", RowBox[List["1", "-", "x"]]], RowBox[List[RowBox[List[SubsuperscriptBox["zz", "1", RowBox[List[SubscriptBox["bb", "1"], "-", "1"]]], " ", SubsuperscriptBox["zz", "2", RowBox[List[SubscriptBox["bb", "2"], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "x", "-", "y"]], ")"]], RowBox[List["c", "-", SubscriptBox["bb", "1"], "-", SubscriptBox["bb", "2"], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["x", " ", SubscriptBox["zz", "1"]]], "-", RowBox[List["y", " ", SubscriptBox["zz", "2"]]]]], ")"]], RowBox[List["-", "a"]]]]], RowBox[List["\[DifferentialD]", "y"]], RowBox[List["\[DifferentialD]", "x"]]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["c", "-", SubscriptBox["bb", "1"], "-", SubscriptBox["bb", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "2"], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", SubscriptBox["bb", "1"], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", SubscriptBox["bb", "2"], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["c", "-", SubscriptBox["bb", "1"], "-", SubscriptBox["bb", "2"]]], "]"]], ">", "0"]]]]]]]]]]










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





2001-10-29





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