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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.06.0037.01

 Input Form

 Hypergeometric2F1[a, b, c, z] \[Proportional] (Gamma[c]/(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)]) + (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[1 + a] Gamma[1 + b] Hypergeometric2F1Regularized[1 + a, 1 + b, 2 + a + b - c, 1 - Subscript[z, 0]] + MeijerG[{{-a, -b}, {}}, {{0, -1 - 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)])) (z - Subscript[z, 0]) + (1/2) (-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[2 + a] Gamma[2 + b] Hypergeometric2F1Regularized[2 + a, 2 + b, 3 + a + b - c, 1 - Subscript[z, 0]] + MeijerG[{{-1 - a, -1 - b}, {}}, {{0, -2 - 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)])) (z - Subscript[z, 0])^2 + O[(z - Subscript[z, 0])^3])

 Standard Form

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 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 ) 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] + ( G 2 , 2 2 , 2 ( 1 - z 0 - a , - b 0 , - a - b + c - 1 ) 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["-", "a"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "b"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", "c", "-", "1"]], 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 + 1 ) Γ ( b + 1 ) 2 F ~ 1 ( a + 1 , b + 1 ; a + b - c + 2 ; 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[RowBox[List["a", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "b", "-", "c", "+", "2"]], 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] ) ( z - z 0 ) + 1 2 ( G 2 , 2 2 , 2 ( 1 - z 0 - a - 1 , - b - 1 0 , - a - b + c - 2 ) 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[RowBox[List["-", "a"]], "-", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "b"]], "-", "1"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", "c", "-", "2"]], 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 + 2 ) Γ ( b + 2 ) 2 F ~ 1 ( a + 2 , b + 2 ; a + b - c + 3 ; 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[RowBox[List["a", "+", "2"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "2"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "b", "-", "c", "+", "3"]], 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] ) ( z - z 0 ) 2 + O ( ( z - z 0 ) 3 ) ) Proportional Hypergeometric2F1 a b c z Gamma c 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 MeijerG -1 a -1 b 0 -1 a -1 b c -1 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 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 1 Gamma b 1 Hypergeometric2F1Regularized a 1 b 1 a b -1 c 2 1 -1 Subscript z 0 z -1 Subscript z 0 1 2 MeijerG -1 a -1 -1 b -1 0 -1 a -1 b c -2 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 2 Gamma b 2 Hypergeometric2F1Regularized a 2 b 2 a b -1 c 3 1 -1 Subscript z 0 z -1 Subscript z 0 2 O z -1 Subscript z 0 3 [/itex]

 Rule Form

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

 2007-05-02