html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 InverseGammaRegularized

 http://functions.wolfram.com/06.12.20.0002.01

 Input Form

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

 Standard Form

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 MathML Form

 2 Q GammaRegularized - 1 ( a , z ) a 2 2 2 w ( 1 - a + w ) Γ ( a ) 2 2 F ~ 2 ( a , a ; a + 1 , a + 1 ; - w ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", "w"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ( Γ ( a ) ( ( z - 1 ) log ( w ) + ψ TagBox["\[Psi]", PolyGamma] ( a ) ) - Γ ( a , w ) ψ TagBox["\[Psi]", PolyGamma] ( a ) ) w 1 - a + w ( w ( 1 - a + w ) Γ ( a ) 2 ( ( z - 1 ) log ( w ) + ψ TagBox["\[Psi]", PolyGamma] ( a ) ) 2 + Γ ( a , w ) ( ( log 2 ( w ) - ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( a ) ) w a + ( w ( 1 - a + w ) Γ ( a , w ) - w a ) ψ TagBox["\[Psi]", PolyGamma] ( a ) 2 + 2 a w Γ ( a ) ψ TagBox["\[Psi]", PolyGamma] ( a ) ( ( z - 1 ) log ( w ) + ψ TagBox["\[Psi]", PolyGamma] ( a ) ) ) + Γ ( a ) ( ( ( 1 - 2 z ) log 2 ( w ) + ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( a ) ) w a + ( w a - 2 w ( w + 1 ) Γ ( a , w ) ) ψ TagBox["\[Psi]", PolyGamma] ( a ) 2 + 2 ( z - 1 ) ( w a - w ( w + 1 ) Γ ( a , w ) ) log ( w ) ψ TagBox["\[Psi]", PolyGamma] ( a ) ) ) w 1 - 2 a + w Γ ( a ) 3 w ( w ( 1 - a + w ) Γ ( a ) 2 F ~ 2 ( a , a ; a + 1 , a + 1 ; - w ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", "w"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] 2 - 2 3 F ~ 3 ( a , a , a ; a + 1 , a + 1 , a + 1 ; - w ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["a", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", "w"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) /; w Q GammaRegularized - 1 ( a , z ) Condition a 2 InverseGammaRegularized a z 2 2 w 1 -1 a w Gamma a 2 HypergeometricPFQRegularized a a a 1 a 1 -1 w Gamma a z -1 w PolyGamma a -1 Gamma a w PolyGamma a w 1 -1 a w w 1 -1 a w Gamma a 2 z -1 w PolyGamma a 2 Gamma a w w 2 -1 PolyGamma 1 a w a w 1 -1 a w Gamma a w -1 w a PolyGamma a 2 2 a w Gamma a PolyGamma a z -1 w PolyGamma a Gamma a 1 -1 2 z w 2 PolyGamma 1 a w a w a -1 2 w w 1 Gamma a w PolyGamma a 2 2 z -1 w a -1 w w 1 Gamma a w w PolyGamma a w 1 -1 2 a w Gamma a 3 w w 1 -1 a w Gamma a HypergeometricPFQRegularized a a a 1 a 1 -1 w 2 -1 2 HypergeometricPFQRegularized a a a a 1 a 1 a 1 -1 w w InverseGammaRegularized a z [/itex]

 Rule Form

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

 2001-10-29