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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.06.0040.01

 Input Form

 Hypergeometric2F1Regularized[a, b, c, z] == (Pi/(Gamma[a] Gamma[b] Gamma[c - a] Gamma[c - b])) Sum[((-1)^k/k!) (Gamma[a + k] Gamma[b + k] (-2 I E^(I (c - a - b) Pi Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)]) Floor[(Pi + Arg[1 - Subscript[z, 0]])/(2 Pi)] Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)] + Csc[(c - a - b) Pi] (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)])) Hypergeometric2F1Regularized[a + k, b + k, 1 + a + b - c + k, 1 - Subscript[z, 0]] - Gamma[-a + c] Gamma[-b + c] (1 - Subscript[z, 0])^(-a - b + c - k) Csc[(c - a - b) Pi] (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)]) Hypergeometric2F1Regularized[-a + c, -b + c, 1 - a - b + c - k, 1 - Subscript[z, 0]]) (z - Subscript[z, 0])^k, {k, 0, Infinity}] /; !Element[c - a - b, Integers]

 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[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] π Γ ( a ) Γ ( b ) Γ ( c - a ) Γ ( c - b ) k = 0 ( - 1 ) k k ! ( Γ ( a + k ) Γ ( b + k ) ( csc ( ( c - a - b ) π ) ( 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 arg ( 1 - z 0 ) + π 2 π arg ( z 0 - z ) 2 π ) 2 F ~ 1 ( a + k , b + k ; a + b - c + k + 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[RowBox[List["a", "+", "k"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "k"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "b", "-", "c", "+", "k", "+", "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] - Γ ( c - a ) Γ ( c - b ) ( 1 - z 0 ) c - a - b - k csc ( ( c - a - b ) π ) ( 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 F ~ 1 ( c - a , c - b ; - a - b + c - k + 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[RowBox[List["c", "-", "a"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["c", "-", "b"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "b", "+", "c", "-", "k", "+", "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] ) ( z - z 0 ) k /; c - a - b TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Hypergeometric2F1Regularized a b c z Gamma a Gamma b Gamma c -1 a Gamma c -1 b -1 k 0 -1 k k -1 Gamma a k Gamma b k c -1 a -1 b 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 Hypergeometric2F1Regularized a k b k a b -1 c k 1 1 -1 Subscript z 0 -1 Gamma c -1 a Gamma c -1 b 1 -1 Subscript z 0 c -1 a -1 b -1 k c -1 a -1 b 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 Hypergeometric2F1Regularized c -1 a c -1 b -1 a -1 b c -1 k 1 1 -1 Subscript z 0 z -1 Subscript z 0 k c -1 a -1 b [/itex]

 Rule Form

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

 2007-05-02