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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==infinity > For the function itself





http://functions.wolfram.com/07.36.06.0016.02









  


  








Input Form





AppellF1[a, Subscript[b, 1], Subscript[b, 2], c, Subscript[z, 1], Subscript[z, 2]] == (((Gamma[c] Gamma[Subscript[b, 1] - a])/(Gamma[c - a] Gamma[Subscript[b, 1]])) AppellF1[a, Subscript[b, 2], 1 + a - c, 1 + a - Subscript[b, 1], Subscript[z, 2]/Subscript[z, 1], 1/Subscript[z, 1]])/(-Subscript[z, 1])^a + (((Gamma[c] Gamma[a - Subscript[b, 1]])/ (Gamma[a] Gamma[c - Subscript[b, 1]])) ((((a - Subscript[b, 1])^2 Subscript[b, 2])/(c - Subscript[b, 1])^2) Subscript[z, 2] HypergeometricPFQ[{{a - Subscript[b, 1] + 1, Subscript[b, 2] + 1}, {Subscript[b, 1]}, {a - Subscript[b, 1] + 1, 1}}, {{c - Subscript[b, 1] + 1, 2}, {}, {c - Subscript[b, 1] + 1}}, Subscript[z, 2]/Subscript[z, 1], Subscript[z, 2]] + HypergeometricPFQ[{{Subscript[b, 1]}, {Subscript[b, 2], a - Subscript[b, 1]}, {1 - c + Subscript[b, 1], 1}}, {{1}, {c - Subscript[b, 1]}, {1 - a + Subscript[b, 1]}}, Subscript[z, 2]/Subscript[z, 1], 1/Subscript[z, 1]]))/ (-Subscript[z, 1])^Subscript[b, 1] /; !Element[a - Subscript[b, 1], Integers]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "c", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "1"], "-", "a"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["c", "-", "a"]], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["z", "1"]]], ")"]], RowBox[List["-", "a"]]], RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["b", "2"], ",", RowBox[List["1", "+", "a", "-", "c"]], ",", RowBox[List["1", "+", "a", "-", SubscriptBox["b", "1"]]], ",", FractionBox[SubscriptBox["z", "2"], SubscriptBox["z", "1"]], ",", FractionBox["1", SubscriptBox["z", "1"]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], RowBox[List["Gamma", "[", RowBox[List["a", "-", SubscriptBox["b", "1"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", SubscriptBox["b", "1"]]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["z", "1"]]], ")"]], RowBox[List["-", SubscriptBox["b", "1"]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["b", "1"]]], ")"]], "2"], SubscriptBox["b", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", SubscriptBox["b", "1"]]], ")"]], "2"]], SubscriptBox["z", "2"], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["a", "-", SubscriptBox["b", "1"], "+", "1"]], ",", RowBox[List[SubscriptBox["b", "2"], "+", "1"]]]], "}"]], ",", RowBox[List["{", SubscriptBox["b", "1"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a", "-", SubscriptBox["b", "1"], "+", "1"]], ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["c", "-", SubscriptBox["b", "1"], "+", "1"]], ",", "2"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["c", "-", SubscriptBox["b", "1"], "+", "1"]], "}"]]]], "}"]], ",", FractionBox[SubscriptBox["z", "2"], SubscriptBox["z", "1"]], ",", SubscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", SubscriptBox["b", "1"], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "2"], ",", RowBox[List["a", "-", SubscriptBox["b", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "c", "+", SubscriptBox["b", "1"]]], ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["c", "-", SubscriptBox["b", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "a", "+", SubscriptBox["b", "1"]]], "}"]]]], "}"]], ",", FractionBox[SubscriptBox["z", "2"], SubscriptBox["z", "1"]], ",", FractionBox["1", SubscriptBox["z", "1"]]]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["a", "-", SubscriptBox["b", "1"]]], ",", "Integers"]], "]"]], "]"]]]]]]








MathML Form







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</ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> - </ms> <ms> c </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 1 </ms> </apply> </apply> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 1 </ms> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <ms> c </ms> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <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> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <ms> a </ms> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> c </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> F </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> 1 </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> 2 </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> [ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ms> ; </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 2 </ms> </apply> </list> </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> 1 </ms> </apply> <ms> - </ms> <ms> c </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> ; </ms> </list> </apply> </list> </apply> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> c </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> 1 </ms> </apply> <ms> - </ms> <ms> a </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 1 </ms> </apply> </apply> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 1 </ms> </apply> </apply> <ms> ] </ms> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <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> </list> </apply> <ms> 2 </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 2 </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> c </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> F </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> 0 </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> 1 </ms> <ms> 2 </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> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 2 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ; </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> ; </ms> </list> </apply> </list> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> c </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> ; </ms> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> c </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 1 </ms> </apply> </apply> </list> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </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> &#8713; </ms> <apply> <ci> TagBox </ci> <ms> &#8484; </ms> <apply> <ci> Function </ci> <integers /> </apply> </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]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "1"], "-", "a"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["zz", "1"]]], ")"]], RowBox[List["-", "a"]]], " ", RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["bb", "2"], ",", RowBox[List["1", "+", "a", "-", "c"]], ",", RowBox[List["1", "+", "a", "-", SubscriptBox["bb", "1"]]], ",", FractionBox[SubscriptBox["zz", "2"], SubscriptBox["zz", "1"]], ",", FractionBox["1", SubscriptBox["zz", "1"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["c", "-", "a"]], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "1"], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "-", SubscriptBox["bb", "1"]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["zz", "1"]]], ")"]], RowBox[List["-", SubscriptBox["bb", "1"]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["bb", "1"]]], ")"]], "2"], " ", SubscriptBox["bb", "2"]]], ")"]], " ", SubscriptBox["zz", "2"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["a", "-", SubscriptBox["bb", "1"], "+", "1"]], ",", RowBox[List[SubscriptBox["bb", "2"], "+", "1"]]]], "}"]], ",", RowBox[List["{", SubscriptBox["bb", "1"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a", "-", SubscriptBox["bb", "1"], "+", "1"]], ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["c", "-", SubscriptBox["bb", "1"], "+", "1"]], ",", "2"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["c", "-", SubscriptBox["bb", "1"], "+", "1"]], "}"]]]], "}"]], ",", FractionBox[SubscriptBox["zz", "2"], SubscriptBox["zz", "1"]], ",", SubscriptBox["zz", "2"]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", SubscriptBox["bb", "1"]]], ")"]], "2"]], "+", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", SubscriptBox["bb", "1"], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "2"], ",", RowBox[List["a", "-", SubscriptBox["bb", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "c", "+", SubscriptBox["bb", "1"]]], ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["c", "-", SubscriptBox["bb", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "a", "+", SubscriptBox["bb", "1"]]], "}"]]]], "}"]], ",", FractionBox[SubscriptBox["zz", "2"], SubscriptBox["zz", "1"]], ",", FractionBox["1", SubscriptBox["zz", "1"]]]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", SubscriptBox["bb", "1"]]], "]"]]]]]]], "/;", RowBox[List["!", RowBox[List[RowBox[List["a", "-", SubscriptBox["bb", "1"]]], "\[Element]", "Integers"]]]]]]]]]]


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





2001-10-29