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http://functions.wolfram.com/06.23.06.0007.01
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InverseBetaRegularized[z, a, b] \[Proportional]
InverseBetaRegularized[z, a, Subscript[b, 0]] -
(1 - w)^(1 - Subscript[b, 0]) w^(1 - a)
(((1 - w)^Subscript[b, 0]/Subscript[b, 0]^2) HypergeometricPFQ[
{1 - a, Subscript[b, 0], Subscript[b, 0]}, {1 + Subscript[b, 0],
1 + Subscript[b, 0]}, 1 - w] - Beta[1 - w, Subscript[b, 0], a]
(Log[1 - w] - PolyGamma[Subscript[b, 0]] +
PolyGamma[a + Subscript[b, 0]])) (b - Subscript[b, 0]) +
(1/2) w^(1 - 2 a) (-2 (-1 + w) w^a Gamma[Subscript[b, 0]]^3
HypergeometricPFQRegularized[{Subscript[b, 0], Subscript[b, 0],
Subscript[b, 0], 1 - a}, {1 + Subscript[b, 0], 1 + Subscript[b, 0],
1 + Subscript[b, 0]}, 1 - w] + (1 - w)^(1 - Subscript[b, 0]) w^a
Beta[1 - w, Subscript[b, 0], a] (PolyGamma[Subscript[b, 0]] -
PolyGamma[a + Subscript[b, 0]]) (Log[1 - w] -
PolyGamma[Subscript[b, 0]] + PolyGamma[a + Subscript[b, 0]]) +
(1 - a) (1 - w)^(2 - Subscript[b, 0]) Gamma[Subscript[b, 0]]^2
HypergeometricPFQRegularized[{Subscript[b, 0], Subscript[b, 0], 1 - a},
{1 + Subscript[b, 0], 1 + Subscript[b, 0]}, 1 - w]
((1 - w)^Subscript[b, 0] Gamma[Subscript[b, 0]]^2
HypergeometricPFQRegularized[{Subscript[b, 0], Subscript[b, 0],
1 - a}, {1 + Subscript[b, 0], 1 + Subscript[b, 0]}, 1 - w] -
Beta[1 - w, Subscript[b, 0], a] (Log[1 - w] -
PolyGamma[Subscript[b, 0]] + PolyGamma[a + Subscript[b, 0]])) -
(1 - w)^(2 - Subscript[b, 0]) w Gamma[Subscript[b, 0]]^2
HypergeometricPFQRegularized[{Subscript[b, 0], Subscript[b, 0], 1 - a},
{1 + Subscript[b, 0], 1 + Subscript[b, 0]}, 1 - w]
((1 - w)^Subscript[b, 0] Gamma[Subscript[b, 0]]^2
HypergeometricPFQRegularized[{Subscript[b, 0], Subscript[b, 0],
1 - a}, {1 + Subscript[b, 0], 1 + Subscript[b, 0]}, 1 - w] -
Beta[1 - w, Subscript[b, 0], a] (Log[1 - w] -
PolyGamma[Subscript[b, 0]] + PolyGamma[a + Subscript[b, 0]])) +
(-1 + a) (1 - w)^(2 - 2 Subscript[b, 0]) Beta[1 - w, Subscript[b, 0], a]
(Log[1 - w] - PolyGamma[Subscript[b, 0]] +
PolyGamma[a + Subscript[b, 0]]) ((1 - w)^Subscript[b, 0]
Gamma[Subscript[b, 0]]^2 HypergeometricPFQRegularized[
{Subscript[b, 0], Subscript[b, 0], 1 - a}, {1 + Subscript[b, 0],
1 + Subscript[b, 0]}, 1 - w] - Beta[1 - w, Subscript[b, 0], a]
(Log[1 - w] - PolyGamma[Subscript[b, 0]] +
PolyGamma[a + Subscript[b, 0]])) +
((-1 + w) Beta[1 - w, Subscript[b, 0], a] (Log[1 - w] -
PolyGamma[Subscript[b, 0]] + PolyGamma[a + Subscript[b, 0]])
((-1 + Subscript[b, 0]) (1 - w)^Subscript[b, 0] w
Gamma[Subscript[b, 0]]^2 HypergeometricPFQRegularized[
{Subscript[b, 0], Subscript[b, 0], 1 - a}, {1 + Subscript[b, 0],
1 + Subscript[b, 0]}, 1 - w] + (1 - w)^Subscript[b, 0] w^a
Log[1 - w] - (-1 + Subscript[b, 0]) w Beta[1 - w, Subscript[b, 0],
a] (Log[1 - w] - PolyGamma[Subscript[b, 0]] +
PolyGamma[a + Subscript[b, 0]])))/(1 - w)^(2 Subscript[b, 0]) +
((-1 + w) Beta[1 - w, Subscript[b, 0], a]
((-(1 - w)^Subscript[b, 0]) w Gamma[Subscript[b, 0]]^2
HypergeometricPFQRegularized[{Subscript[b, 0], Subscript[b, 0],
1 - a}, {1 + Subscript[b, 0], 1 + Subscript[b, 0]}, 1 - w] +
w Beta[1 - w, Subscript[b, 0], a] (Log[1 - w] -
PolyGamma[Subscript[b, 0]] + PolyGamma[a + Subscript[b, 0]]) +
(1 - w)^Subscript[b, 0] w^a (PolyGamma[1, Subscript[b, 0]] -
PolyGamma[1, a + Subscript[b, 0]])))/(1 - w)^(2 Subscript[b, 0]))
(b - Subscript[b, 0])^2 + \[Ellipsis] /; (b -> Subscript[b, 0]) &&
w == InverseBetaRegularized[z, a, Subscript[b, 0]]
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<msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> b </mi> <mn> 0 </mn> </msub> </mrow> </mrow> </msup> <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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> ψ </mi> <annotation 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</mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <msub> <mi> b </mi> <mn> 0 </mn> </msub> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> w </mi> <mi> a </mi> </msup> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> w </mi> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <msub> <mi> b </mi> <mn> 0 </mn> </msub> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <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[SubscriptBox["b", "0"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "0"], 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</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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> </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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> </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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> </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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <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["(", 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</mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> w </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <msub> <mi> b </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mtext> </mtext> <mo> + </mo> <mo> … </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <msub> <mi> b </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </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> <msub> <mi> b </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> 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Date Added to functions.wolfram.com (modification date)
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){kind=small}
