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AppellF1






Mathematica Notation

Traditional Notation









Hypergeometric Functions > AppellF1[a,b1,b2,c,z1,z2] > Series representations > Generalized power series > Expansions at (z1,z2)==(0,0) > For the function itself





http://functions.wolfram.com/07.36.06.0001.01









  


  










Input Form





AppellF1[a, Subscript[b, 1], Subscript[b, 2], c, Subscript[z, 1], Subscript[z, 2]] \[Proportional] 1 + (a Subscript[b, 1] Subscript[z, 1])/ c + (a (1 + a) Subscript[b, 1] (1 + Subscript[b, 1]) Subscript[z, 1]^2)/ (2 c (1 + c)) + (a Subscript[b, 2] Subscript[z, 2])/c + (a (1 + a) Subscript[b, 1] Subscript[b, 2] Subscript[z, 1] Subscript[z, 2])/(c (1 + c)) + (a (1 + a) (2 + a) Subscript[b, 1] (1 + Subscript[b, 1]) Subscript[b, 2] Subscript[z, 1]^2 Subscript[z, 2])/ (2 c (1 + c) (2 + c)) + (a (1 + a) Subscript[b, 2] (1 + Subscript[b, 2]) Subscript[z, 2]^2)/(2 c (1 + c)) + (a (1 + a) (2 + a) Subscript[b, 1] Subscript[b, 2] (1 + Subscript[b, 2]) Subscript[z, 1] Subscript[z, 2]^2)/(2 c (1 + c) (2 + c)) + (a (1 + a) (2 + a) (3 + a) Subscript[b, 1] (1 + Subscript[b, 1]) Subscript[b, 2] (1 + Subscript[b, 2]) Subscript[z, 1]^2 Subscript[z, 2]^2)/(4 c (1 + c) (2 + c) (3 + c)) + \[Ellipsis] /; (Subscript[z, 1] -> 0) && (Subscript[z, 2] -> 0)










Standard Form





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MathML Form







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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> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> a </ci> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> a </ci> </apply> <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> <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> <power /> <apply> <times /> <ci> c </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> a </ci> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> c </ci> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> 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type='integer'> 1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> c </ci> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> a </ci> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <plus /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> c </ci> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <plus /> <cn type='integer'> 3 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Rule </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </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[RowBox[List["1", "+", FractionBox[RowBox[List["a", " ", SubscriptBox["bb", "1"], " ", SubscriptBox["zz", "1"]]], "c"], "+", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]], " ", SubscriptBox["bb", "1"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["bb", "1"]]], ")"]], " ", SubsuperscriptBox["zz", "1", "2"]]], RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List["1", "+", "c"]], ")"]]]]], "+", FractionBox[RowBox[List["a", " ", SubscriptBox["bb", "2"], " ", SubscriptBox["zz", "2"]]], "c"], "+", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]], " ", SubscriptBox["bb", "1"], " ", SubscriptBox["bb", "2"], " ", SubscriptBox["zz", "1"], " ", SubscriptBox["zz", "2"]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", "c"]], ")"]]]]], "+", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a"]], ")"]], " ", SubscriptBox["bb", "1"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["bb", "1"]]], ")"]], " ", SubscriptBox["bb", "2"], " ", SubsuperscriptBox["zz", "1", "2"], " ", SubscriptBox["zz", "2"]]], RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List["1", "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "c"]], ")"]]]]], "+", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]], " ", SubscriptBox["bb", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["bb", "2"]]], ")"]], " ", SubsuperscriptBox["zz", "2", "2"]]], RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List["1", "+", "c"]], ")"]]]]], "+", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a"]], ")"]], " ", SubscriptBox["bb", "1"], " ", SubscriptBox["bb", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["bb", "2"]]], ")"]], " ", SubscriptBox["zz", "1"], " ", SubsuperscriptBox["zz", "2", "2"]]], RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List["1", "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "c"]], ")"]]]]], "+", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "a"]], ")"]], " ", SubscriptBox["bb", "1"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["bb", "1"]]], ")"]], " ", SubscriptBox["bb", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["bb", "2"]]], ")"]], " ", SubsuperscriptBox["zz", "1", "2"], " ", SubsuperscriptBox["zz", "2", "2"]]], RowBox[List["4", " ", "c", " ", RowBox[List["(", RowBox[List["1", "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "c"]], ")"]]]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["zz", "1"], "\[Rule]", "0"]], ")"]], "&&", RowBox[List["(", RowBox[List[SubscriptBox["zz", "2"], "\[Rule]", "0"]], ")"]]]]]]]]]]










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





2001-10-29





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