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 Beta

 http://functions.wolfram.com/06.19.06.0069.01

 Input Form

 Beta[z, a, b] \[Proportional] Beta[z, a, Subscript[b, 0]] + ((1 - z)^Subscript[b, 0] Gamma[Subscript[b, 0]]^2 HypergeometricPFQRegularized[{1 - a, Subscript[b, 0], Subscript[b, 0]}, {1 + Subscript[b, 0], 1 + Subscript[b, 0]}, 1 - z] - Beta[1 - z, Subscript[b, 0], a] Log[1 - z] + Beta[a, Subscript[b, 0]] (PolyGamma[Subscript[b, 0]] - PolyGamma[a + Subscript[b, 0]])) (b - Subscript[b, 0]) + (1/2) (((2 (1 - z)^Subscript[b, 0])/Subscript[b, 0]^3) (-HypergeometricPFQ[{1 - a, Subscript[b, 0], Subscript[b, 0], Subscript[b, 0]}, {1 + Subscript[b, 0], 1 + Subscript[b, 0], 1 + Subscript[b, 0]}, 1 - z] + Subscript[b, 0] Log[1 - z] HypergeometricPFQ[{1 - a, Subscript[b, 0], Subscript[b, 0]}, {1 + Subscript[b, 0], 1 + Subscript[b, 0]}, 1 - z]) + Beta[a, Subscript[b, 0]] ((PolyGamma[Subscript[b, 0]] - PolyGamma[a + Subscript[b, 0]])^2 + PolyGamma[1, Subscript[b, 0]] - PolyGamma[1, a + Subscript[b, 0]]) - Beta[1 - z, Subscript[b, 0], a] Log[1 - z]^2) (b - Subscript[b, 0])^2 + \[Ellipsis] /; (b -> Subscript[b, 0])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Beta", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["Beta", "[", RowBox[List["z", ",", "a", ",", SubscriptBox["b", "0"]]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], SubscriptBox["b", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["b", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "a"]], ",", SubscriptBox["b", "0"], ",", SubscriptBox["b", "0"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["b", "0"]]]]], "}"]], ",", RowBox[List["1", "-", "z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[RowBox[List["1", "-", "z"]], ",", SubscriptBox["b", "0"], ",", "a"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Beta", "[", RowBox[List["a", ",", SubscriptBox["b", "0"]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", SubscriptBox["b", "0"], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", SubscriptBox["b", "0"]]], "]"]]]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", SubscriptBox["b", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], SubscriptBox["b", "0"]]]], SubsuperscriptBox["b", "0", "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "a"]], ",", SubscriptBox["b", "0"], ",", SubscriptBox["b", "0"], ",", SubscriptBox["b", "0"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["b", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["b", "0"]]]]], "}"]], ",", RowBox[List["1", "-", "z"]]]], "]"]]]], "+", RowBox[List[SubscriptBox["b", "0"], RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "a"]], ",", SubscriptBox["b", "0"], ",", SubscriptBox["b", "0"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["b", "0"]]]]], "}"]], ",", RowBox[List["1", "-", "z"]]]], "]"]]]]]], ")"]]]], "+", " ", RowBox[List[RowBox[List["Beta", "[", RowBox[List["a", ",", SubscriptBox["b", "0"]]], "]"]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", SubscriptBox["b", "0"], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", SubscriptBox["b", "0"]]], "]"]]]], ")"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", SubscriptBox["b", "0"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["a", "+", SubscriptBox["b", "0"]]]]], "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[RowBox[List["1", "-", "z"]], ",", SubscriptBox["b", "0"], ",", "a"]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], "2"]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", SubscriptBox["b", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List["(", RowBox[List["b", "\[Rule]", SubscriptBox["b", "0"]]], ")"]]]]]]

 MathML Form

 Β Beta z ( a , b ) Β Beta z ( a , b 0 ) + ( Γ ( b 0 ) 2 3 F ~ 2 ( 1 - a , b 0 , b 0 ; b 0 + 1 , b 0 + 1 ; 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "a"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "0"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "0"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "0"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "0"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", "z"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ( 1 - z ) b 0 - Β Beta 1 - z ( b 0 , a ) log ( 1 - z ) + Β Beta ( a , b 0 ) ( ψ TagBox["\[Psi]", PolyGamma] ( b 0 ) - ψ TagBox["\[Psi]", PolyGamma] ( a + b 0 ) ) ) ( b - b 0 ) + 1 2 ( - Β Beta 1 - z ( b 0 , a ) log 2 ( 1 - z ) + ( ( ψ TagBox["\[Psi]", PolyGamma] ( b 0 ) - ψ TagBox["\[Psi]", PolyGamma] ( a + b 0 ) ) 2 + ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( b 0 ) - ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( a + b 0 ) ) Β Beta ( a , b 0 ) + 2 ( 1 - z ) b 0 b 0 3 ( b 0 log ( 1 - z ) 3 F 2 ( 1 - a , b 0 , b 0 ; b 0 + 1 , b 0 + 1 ; 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "a"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "0"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "0"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "0"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "0"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", "z"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - 4 F 3 ( 1 - a , b 0 , b 0 , b 0 ; b 0 + 1 , b 0 + 1 , b 0 + 1 ; 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "a"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "0"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "0"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "0"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "0"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "0"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "0"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", "z"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) ) ( b - b 0 ) 2 + /; ( b "\[Rule]" b 0 ) Condition Proportional Beta z a b Beta z a Subscript b 0 Gamma Subscript b 0 2 HypergeometricPFQRegularized 1 -1 a Subscript b 0 Subscript b 0 Subscript b 0 1 Subscript b 0 1 1 -1 z 1 -1 z Subscript b 0 -1 Beta 1 -1 z Subscript b 0 a 1 -1 z Beta a Subscript b 0 PolyGamma Subscript b 0 -1 PolyGamma a Subscript b 0 b -1 Subscript b 0 1 2 -1 Beta 1 -1 z Subscript b 0 a 1 -1 z 2 PolyGamma Subscript b 0 -1 PolyGamma a Subscript b 0 2 PolyGamma 1 Subscript b 0 -1 PolyGamma 1 a Subscript b 0 Beta a Subscript b 0 2 1 -1 z Subscript b 0 Subscript b 0 3 -1 Subscript b 0 1 -1 z HypergeometricPFQ 1 -1 a Subscript b 0 Subscript b 0 Subscript b 0 1 Subscript b 0 1 1 -1 z -1 HypergeometricPFQ 1 -1 a Subscript b 0 Subscript b 0 Subscript b 0 Subscript b 0 1 Subscript b 0 1 Subscript b 0 1 1 -1 z b -1 Subscript b 0 2 Rule b Subscript b 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Beta", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Beta", "[", RowBox[List["z", ",", "a", ",", SubscriptBox["bb", "0"]]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], SubscriptBox["bb", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["bb", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "a"]], ",", SubscriptBox["bb", "0"], ",", SubscriptBox["bb", "0"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["bb", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["bb", "0"]]]]], "}"]], ",", RowBox[List["1", "-", "z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[RowBox[List["1", "-", "z"]], ",", SubscriptBox["bb", "0"], ",", "a"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Beta", "[", RowBox[List["a", ",", SubscriptBox["bb", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", SubscriptBox["bb", "0"], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", SubscriptBox["bb", "0"]]], "]"]]]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", SubscriptBox["bb", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], SubscriptBox["bb", "0"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "a"]], ",", SubscriptBox["bb", "0"], ",", SubscriptBox["bb", "0"], ",", SubscriptBox["bb", "0"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["bb", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["bb", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["bb", "0"]]]]], "}"]], ",", RowBox[List["1", "-", "z"]]]], "]"]]]], "+", RowBox[List[SubscriptBox["bb", "0"], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "a"]], ",", SubscriptBox["bb", "0"], ",", SubscriptBox["bb", "0"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["bb", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["bb", "0"]]]]], "}"]], ",", RowBox[List["1", "-", "z"]]]], "]"]]]]]], ")"]]]], SubsuperscriptBox["bb", "0", "3"]], "+", RowBox[List[RowBox[List["Beta", "[", RowBox[List["a", ",", SubscriptBox["bb", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", SubscriptBox["bb", "0"], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", SubscriptBox["bb", "0"]]], "]"]]]], ")"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", SubscriptBox["bb", "0"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["a", "+", SubscriptBox["bb", "0"]]]]], "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[RowBox[List["1", "-", "z"]], ",", SubscriptBox["bb", "0"], ",", "a"]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", SubscriptBox["bb", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["b", "\[Rule]", SubscriptBox["bb", "0"]]], ")"]]]]]]]]

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

 2007-05-02