Inverse of the regularized incomplete beta function: Differentiation (formula 06.23.20.0006)

InverseBetaRegularized

$Mathematica Notation$

Gamma, Beta, Erf

InverseBetaRegularized[z,a,b]

Differentiation

Low-order differentiation

With respect to b

http://functions.wolfram.com/06.23.20.0006.01

Input Form

D[InverseBetaRegularized[z, a, b], {b, 2}] == w^(1 - 2 a) (-2 (-1 + w) w^a Gamma[b]^3 HypergeometricPFQRegularized[ {b, b, b, 1 - a}, {1 + b, 1 + b, 1 + b}, 1 - w] + (1 - w)^(1 - b) w^a Beta[1 - w, b, a] (PolyGamma[b] - PolyGamma[a + b]) (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b]) + (1 - a) (1 - w)^(2 - b) Gamma[b]^2 HypergeometricPFQRegularized[ {b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] ((1 - w)^b Gamma[b]^2 HypergeometricPFQRegularized[{b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] - Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b])) - (1 - w)^(2 - b) w Gamma[b]^2 HypergeometricPFQRegularized[{b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] ((1 - w)^b Gamma[b]^2 HypergeometricPFQRegularized[{b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] - Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b])) + (-1 + a) (1 - w)^(2 - 2 b) Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b]) ((1 - w)^b Gamma[b]^2 HypergeometricPFQRegularized[ {b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] - Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b])) + ((-1 + w) Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b]) ((-1 + b) (1 - w)^b w Gamma[b]^2 HypergeometricPFQRegularized[{b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] + (1 - w)^b w^a Log[1 - w] - (-1 + b) w Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b])))/(1 - w)^(2 b) + ((-1 + w) Beta[1 - w, b, a] ((-(1 - w)^b) w Gamma[b]^2 HypergeometricPFQRegularized[{b, b, 1 - a}, {1 + b, 1 + b}, 1 - w] + w Beta[1 - w, b, a] (Log[1 - w] - PolyGamma[b] + PolyGamma[a + b]) + (1 - w)^b w^a (PolyGamma[1, b] - PolyGamma[1, a + b])))/ (1 - w)^(2 b)) /; w == InverseBetaRegularized[z, a, b]

Standard Form

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MathML Form

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HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", "w"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> <mo> ⁢ </mo> <mi> w </mi> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> Β </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> w </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> Β </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> w </mi> <mi> a </mi> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 3 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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["b", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["b", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "a"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["b", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", "w"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> <mo> ⁢ </mo> <mi> w </mi> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <semantics> <mi> Β </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> w </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> w </mi> <mo> ⩵ </mo> <mrow> <msubsup> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> z </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> b </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> InverseBetaRegularized </ci> <ci> z </ci> <ci> a </ci> <ci> b </ci> </apply> </apply> <apply> <times /> <apply> <power /> <ci> w </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <ci> b </ci> <ci> b </ci> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </list> <list> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <power /> <ci> w </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <ci> Beta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <ci> b </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <ci> w </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <ci> b </ci> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </list> <list> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <ci> b </ci> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </list> <list> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Beta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <ci> b </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> </apply> </apply> </apply> <ci> w </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <ci> b </ci> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </list> <list> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <ci> b </ci> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </list> <list> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Beta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <ci> b </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <ci> Beta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <ci> b </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <ci> b </ci> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </list> <list> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Beta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <ci> b </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -2 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Beta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <ci> b </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <power /> <ci> w </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <ci> b </ci> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </list> <list> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Beta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <ci> b </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -2 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Beta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <ci> b </ci> </apply> <apply> <times /> 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Date Added to functions.wolfram.com (modification date)

2001-10-29

InverseBetaRegularized[z₁,z₂,a,b]