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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.06.0046.01

 Input Form

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

 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 2 csc ( c π ) ( 1 - x ) c - a - b + c Γ ( a ) Γ ( b ) ( c - a - b ) π arg ( x - z ) 2 π k = 0 ( 1 - x ) - k k ! ( 2 x - c - k + 1 ( 1 - x ) a + b - c + k Γ ( c - a ) Γ ( c - b ) arg ( x - z ) 2 π 2 F ~ 1 ( a - c + 1 , b - c + 1 ; - c - k + 2 ; x ) 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", "-", "c", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "c", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "c"]], "-", "k", "+", "2"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox["x", Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] + ( csc ( ( a + b - c ) π ) Γ ( 1 - a - k ) Γ ( 1 - b - k ) ( - a - b + c ) π arg ( x - z ) 2 π - sin ( ( c - a ) π ) sin ( ( c - b ) π ) Γ ( a + k ) Γ ( b + k ) π 2 ( ( c - a - b ) π arg ( x - z ) 2 π csc ( ( a + b - c ) π ) + 2 arg ( x - z ) 2 π ) ) 2 F ~ 1 ( c - a , c - b ; c + k ; x ) 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["c", "+", "k"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox["x", Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] ) ( z - x ) k /; c - a - b TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] c TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x > 1 Condition Hypergeometric2F1Regularized a b c z 2 c 1 -1 x c -1 a -1 b c Gamma a Gamma b -1 c -1 a -1 b x -1 z 2 -1 k 0 1 -1 x -1 k k -1 2 x -1 c -1 k 1 1 -1 x a b -1 c k Gamma c -1 a Gamma c -1 b -1 x -1 z 2 -1 Hypergeometric2F1Regularized a -1 c 1 b -1 c 1 -1 c -1 k 2 x a b -1 c Gamma 1 -1 a -1 k Gamma 1 -1 b -1 k -1 -1 a -1 b c x -1 z 2 -1 -1 c -1 a c -1 b Gamma a k Gamma b k 2 -1 c -1 a -1 b x -1 z 2 -1 a b -1 c 2 x -1 z 2 -1 Hypergeometric2F1Regularized c -1 a c -1 b c k x z -1 x k c -1 a -1 b c x x 1 [/itex]

 Rule Form

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

 2007-05-02