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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.06.0041.01

 Input Form

 Hypergeometric2F1Regularized[a, b, c, z] \[Proportional] (1/(Gamma[a] Gamma[b] Gamma[-a + c] Gamma[-b + c])) (-2 I E^(I (c - a - b) Pi Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)]) Pi Floor[(Pi + Arg[1 - Subscript[z, 0]])/(2 Pi)] Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)] Gamma[a] Gamma[b] Hypergeometric2F1Regularized[a, b, 1 + a + b - c, 1 - Subscript[z, 0]] + MeijerG[{{1 - a, 1 - b}, {}}, {{0, -a - b + c}, {}}, 1 - Subscript[z, 0]] (1/(1 - Subscript[z, 0]))^((c - a - b) Floor[Arg[-z + Subscript[z, 0]]/ (2 Pi)]) (1 - Subscript[z, 0])^((c - a - b) Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)]) + O[z - Subscript[z, 0]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", "a", "-", "b"]], ")"]], " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "-", SubscriptBox["z", "0"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", RowBox[List["1", "+", "a", "+", "b", "-", "c"]], ",", RowBox[List["1", "-", SubscriptBox["z", "0"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "a"]], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", "c"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["1", "-", SubscriptBox["z", "0"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", SubscriptBox["z", "0"]]]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["c", "-", "a", "-", "b"]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["z", "0"]]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["c", "-", "a", "-", "b"]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]], "+", RowBox[List["O", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]]]], ")"]]]]]]]]

 MathML Form

 2 F ~ 1 ( a , b ; c ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] 1 Γ ( a ) Γ ( b ) Γ ( c - a ) Γ ( c - b ) ( G 2 , 2 2 , 2 ( 1 - z 0 1 - a , 1 - b 0 , - a - b + c ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["z", "0"]]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", "a"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "b"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", "c"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ( 1 1 - z 0 ) ( c - a - b ) arg ( z 0 - z ) 2 π ( 1 - z 0 ) ( c - a - b ) arg ( z 0 - z ) 2 π - 2 ( c - a - b ) π arg ( z 0 - z ) 2 π π arg ( 1 - z 0 ) + π 2 π arg ( z 0 - z ) 2 π Γ ( a ) Γ ( b ) 2 F ~ 1 ( a , b ; a + b - c + 1 ; 1 - z 0 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "b", "-", "c", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", SubscriptBox["z", "0"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] + O ( z - z 0 ) ) Proportional Hypergeometric2F1Regularized a b c z 1 Gamma a Gamma b Gamma c -1 a Gamma c -1 b -1 MeijerG 1 -1 a 1 -1 b 0 -1 a -1 b c 1 -1 Subscript z 0 1 1 -1 Subscript z 0 -1 c -1 a -1 b Subscript z 0 -1 z 2 -1 1 -1 Subscript z 0 c -1 a -1 b Subscript z 0 -1 z 2 -1 -1 2 c -1 a -1 b Subscript z 0 -1 z 2 -1 1 -1 Subscript z 0 2 -1 Subscript z 0 -1 z 2 -1 Gamma a Gamma b Hypergeometric2F1Regularized a b a b -1 c 1 1 -1 Subscript z 0 O z -1 Subscript z 0 [/itex]

 Rule Form

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

 2007-05-02