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 BetaRegularized

 http://functions.wolfram.com/06.21.06.0024.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "b"]], "]"]], RowBox[List["Gamma", "[", "a", "]"]]], SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["z", "0"]], ")"]], RowBox[List["a", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["z", "0", RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "a"]], ",", RowBox[List["k", "-", "j"]]]], "]"]], " ", SubsuperscriptBox["z", "0", RowBox[List["j", "-", "k"]]]]], RowBox[List[RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "+", "j"]], "]"]], " ", SubsuperscriptBox["z", "0", RowBox[List[RowBox[List["-", "a"]], "-", "j"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "b", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[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[RowBox[List["Csc", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", SubscriptBox["z", "0"]]]], ")"]], RowBox[List["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["b", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]], ")"]]]], " ", "-", RowBox[List[RowBox[List["Csc", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", RowBox[List["a", "+", "b"]], ",", RowBox[List["1", "+", "b", "-", "j"]], ",", RowBox[List["1", "-", SubscriptBox["z", "0"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", SubscriptBox["z", "0"]]]], ")"]], RowBox[List["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["b", "-", "j", "+", RowBox[List["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["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["b", ",", "Integers"]], "]"]], "]"]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BetaRegularized", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "b"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["zz", "0"]], ")"]], RowBox[List["a", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["zz", "0", RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "a"]], ",", RowBox[List["k", "-", "j"]]]], "]"]], " ", SubsuperscriptBox["zz", "0", RowBox[List["j", "-", "k"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "+", "j"]], "]"]], " ", SubsuperscriptBox["zz", "0", RowBox[List[RowBox[List["-", "a"]], "-", "j"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "b", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "-", SubscriptBox["zz", "0"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[RowBox[List["Csc", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", SubscriptBox["zz", "0"]]]], ")"]], RowBox[List["b", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["zz", "0"]]], ")"]], RowBox[List["b", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]], ")"]]]], "-", RowBox[List[RowBox[List["Csc", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", RowBox[List["a", "+", "b"]], ",", RowBox[List["1", "+", "b", "-", "j"]], ",", RowBox[List["1", "-", SubscriptBox["zz", "0"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", SubscriptBox["zz", "0"]]]], ")"]], RowBox[List["b", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["zz", "0"]]], ")"]], RowBox[List["b", "-", "j", "+", RowBox[List["b", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]]]]]]]]]], RowBox[List["Gamma", "[", "a", "]"]]], "/;", RowBox[List["!", RowBox[List["b", "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02