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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.06.0058.01

 Input Form

 Hypergeometric2F1Regularized[a, b, c, z] \[Proportional] (Gamma[c - a - b]/(Gamma[c - a] Gamma[c - b])) (1 - (a b (z - 1))/(1 + a + b - c) + (a (1 + a) b (1 + b) (z - 1)^2)/ (2 (1 + a + b - c) (2 + a + b - c)) + O[(z - 1)^3]) + (Gamma[a + b - c]/(Gamma[a] Gamma[b])) (1 - z)^(c - a - b) (1 + ((a - c) (-b + c) (z - 1))/(1 - a - b + c) + ((-a + c) (1 - a + c) (-b + c) (1 - b + c) (z - 1)^2)/ (2 (1 - a - b + c) (2 - a - b + c)) + O[(z - 1)^3]) /; (z -> 1) && !Element[c - a - b, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["c", "-", "a", "-", "b"]], "]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["c", "-", "a"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["c", "-", "b"]], "]"]]]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["a", " ", "b", " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], RowBox[List["1", "+", "a", "+", "b", "-", "c"]]], "+", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]], " ", "b", " ", RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "-", "c"]], ")"]]]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "3"], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "-", "c"]], "]"]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["c", "-", "a", "-", "b"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], RowBox[List["1", "-", "a", "-", "b", "+", "c"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", "a", "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", "b", "+", "c"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", "a", "-", "b", "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "-", "a", "-", "b", "+", "c"]], ")"]]]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "3"], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "1"]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["c", "-", "a", "-", "b"]], ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 2 F 1 ( a , b ; c ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] Γ ( c ) Γ ( a + b - c ) Γ ( a ) Γ ( b ) ( 1 - z ) c - a - b ( 1 + ( a - c ) ( - b + c ) ( z - 1 ) 1 - a - b + c + ( - a + c ) ( 1 - a + c ) ( - b + c ) ( 1 - b + c ) ( z - 1 ) 2 2 ( 1 - a - b + c ) ( 2 - a - b + c ) + O ( ( z - 1 ) 3 ) ) + Γ ( c ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) ( 1 - a b ( z - 1 ) 1 + a + b - c + a ( 1 + a ) b ( 1 + b ) ( z - 1 ) 2 2 ( 1 + a + b - c ) ( 2 + a + b - c ) + O ( ( z - 1 ) 3 ) ) /; ( z "\[Rule]" 1 ) c - a - b TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Proportional Hypergeometric2F1 a b c z Gamma c Gamma a b -1 c Gamma a Gamma b -1 1 -1 z c -1 a -1 b 1 a -1 c -1 b c z -1 1 -1 a -1 b c -1 -1 a c 1 -1 a c -1 b c 1 -1 b c z -1 2 2 1 -1 a -1 b c 2 -1 a -1 b c -1 O z -1 3 Gamma c Gamma c -1 a -1 b Gamma c -1 a Gamma c -1 b -1 1 -1 a b z -1 1 a b -1 c -1 a 1 a b 1 b z -1 2 2 1 a b -1 c 2 a b -1 c -1 O z -1 3 Rule z 1 c -1 a -1 b [/itex]

 Rule Form

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

 2007-05-02