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AppellF1




Mathematica Notation

Traditional Notation









Hypergeometric Functions > AppellF1[a,b1,b2,c,z1,z2] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself





http://functions.wolfram.com/07.36.13.0003.01









  


  








Input Form





a Subscript[b, 2] (1 + Subscript[b, 2]) w[Subscript[z, 1], Subscript[z, 2]] + (1 + Subscript[b, 2]) (-c + (2 + 2 a + Subscript[b, 2]) Subscript[z, 2] - (1 + a - Subscript[b, 1]) Subscript[z, 1]) Derivative[0, 1][w][ Subscript[z, 1], Subscript[z, 2]] + ((1 + Subscript[b, 1] + Subscript[b, 2]) (-1 + Subscript[z, 2]) Subscript[z, 2] + (1 + c - Subscript[b, 1]) (-1 + Subscript[z, 2]) (Subscript[z, 2] - Subscript[z, 1]) + (2 + a - c + Subscript[b, 2]) Subscript[z, 2] (Subscript[z, 2] - Subscript[z, 1])) Derivative[0, 2][w][Subscript[z, 1], Subscript[z, 2]] + (-1 + Subscript[z, 2]) Subscript[z, 2] (Subscript[z, 2] - Subscript[z, 1]) Derivative[0, 3][w][Subscript[z, 1], Subscript[z, 2]] == 0 /; w[Subscript[z, 1], Subscript[z, 2]] == AppellF1[a, Subscript[b, 1], Subscript[b, 2], c, Subscript[z, 1], Subscript[z, 2]]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["a", " ", SubscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["b", "2"]]], ")"]], " ", RowBox[List["w", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", RowBox[List["2", " ", "a"]], "+", SubscriptBox["b", "2"]]], ")"]], " ", SubscriptBox["z", "2"]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "-", SubscriptBox["b", "1"]]], ")"]], " ", SubscriptBox["z", "1"]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", RowBox[List["0", ",", "1"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["z", "2"]]], ")"]], " ", SubscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "c", "-", SubscriptBox["b", "1"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[SubscriptBox["z", "2"], "-", SubscriptBox["z", "1"]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "a", "-", "c", "+", SubscriptBox["b", "2"]]], ")"]], " ", SubscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[SubscriptBox["z", "2"], "-", SubscriptBox["z", "1"]]], ")"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", RowBox[List["0", ",", "2"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["z", "2"]]], ")"]], " ", SubscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[SubscriptBox["z", "2"], "-", SubscriptBox["z", "1"]]], ")"]], " ", RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", RowBox[List["0", ",", "3"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]]]], "\[Equal]", "0"]], "/;", " ", RowBox[List[RowBox[List["w", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], "\[Equal]", RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "c", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]]]]]]








MathML Form







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encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ; </mo> <mi> c </mi> <mo> ; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 0 </cn> <cn type='integer'> 3 </cn> </list> <ci> w </ci> </apply> <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> <apply> <times /> <apply> <plus /> <apply> <times /> <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> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <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> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 0 </cn> <cn type='integer'> 2 </cn> </list> <ci> w </ci> </apply> <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> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 0 </cn> <cn type='integer'> 1 </cn> </list> <ci> w </ci> </apply> <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> <apply> <times /> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> b </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> <ci> w </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> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> w </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> <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> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["a_", " ", SubscriptBox["b_", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["b_", "2"]]], ")"]], " ", RowBox[List["w", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["b_", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c_"]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", RowBox[List["2", " ", "a_"]], "+", SubscriptBox["b_", "2"]]], ")"]], " ", SubscriptBox["z_", "2"]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a_", "-", SubscriptBox["b_", "1"]]], ")"]], " ", SubscriptBox["z_", "1"]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", RowBox[List["0", ",", "1"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["b_", "1"], "+", SubscriptBox["b_", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["z_", "2"]]], ")"]], " ", SubscriptBox["z_", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "c_", "-", SubscriptBox["b_", "1"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["z_", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[SubscriptBox["z_", "2"], "-", SubscriptBox["z_", "1"]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "a_", "-", "c_", "+", SubscriptBox["b_", "2"]]], ")"]], " ", SubscriptBox["z_", "2"], " ", RowBox[List["(", RowBox[List[SubscriptBox["z_", "2"], "-", SubscriptBox["z_", "1"]]], ")"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", RowBox[List["0", ",", "2"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["z_", "2"]]], ")"]], " ", SubscriptBox["z_", "2"], " ", RowBox[List["(", RowBox[List[SubscriptBox["z_", "2"], "-", SubscriptBox["z_", "1"]]], ")"]], " ", RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", RowBox[List["0", ",", "3"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], "\[Equal]", RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "c", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]]]]]]]]


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





2001-10-29