Inverse of the regularized incomplete gamma function: Differentiation (formula 06.12.20.0010)

InverseGammaRegularized

$Mathematica Notation$

Gamma, Beta, Erf

InverseGammaRegularized[a,z]

Differentiation

Low-order differentiation

With respect to z

http://functions.wolfram.com/06.12.20.0010.01

Input Form

D[InverseGammaRegularized[a, z], {z, 8}] == E^(8 w) w^(1 - 8 a) (1 - a (28 + a (-322 + a (1960 + a (-6769 + 4 a (3283 + 9 a (-363 + 140 a)))))) + 247 w + a (-3579 + a (21289 + a (-66369 + 4 a (28438 + 45 a (-559 + 196 a))))) w - 6 (-1 + a) (-1 + 2 a) (-947 + a (5891 + 60 a (-206 + 147 a))) w^2 + 2 (-1 + a) (-17729 + 2 a (45001 + 90 a (-853 + 490 a))) w^3 - 20 (-1 + a) (4175 + 9 a (-1343 + 980 a)) w^4 + 36 (-1 + a) (-2323 + 2940 a) w^5 - 35280 (-1 + a) w^6 + 5040 w^7) Gamma[a]^8 /; w == InverseGammaRegularized[a, z]

Standard Form

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MathML Form

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+ </mo> <mn> 1960 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 322 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 28 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mn> 8 </mn> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> w </mi> <mo>  </mo> <mrow> <msup> <semantics> <mi> Q </mi> <annotation-xml encoding='MathML-Content'> <ci> GammaRegularized </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 8 </cn> </degree> </bvar> <apply> <ci> InverseGammaRegularized </ci> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> w </ci> </apply> </apply> <apply> <power /> <ci> w </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 5040 </cn> <apply> <power /> <ci> w </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 35280 </cn> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> w </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 36 </cn> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2940 </cn> <ci> a </ci> </apply> <cn type='integer'> -2323 </cn> </apply> <apply> <power 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</apply> </apply> <cn type='integer'> -17729 </cn> </apply> <apply> <power /> <ci> w </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 60 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 147 </cn> <ci> a </ci> </apply> <cn type='integer'> -206 </cn> </apply> </apply> <cn type='integer'> 5891 </cn> </apply> </apply> <cn type='integer'> -947 </cn> </apply> <apply> <power /> <ci> w </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 45 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 196 </cn> <ci> a </ci> </apply> <cn type='integer'> -559 </cn> </apply> </apply> <cn type='integer'> 28438 </cn> </apply> </apply> <cn type='integer'> -66369 </cn> </apply> </apply> <cn type='integer'> 21289 </cn> </apply> </apply> <cn type='integer'> -3579 </cn> </apply> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> 247 </cn> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 140 </cn> <ci> a </ci> </apply> <cn type='integer'> -363 </cn> </apply> </apply> <cn type='integer'> 3283 </cn> </apply> </apply> <cn type='integer'> -6769 </cn> </apply> </apply> <cn type='integer'> 1960 </cn> </apply> </apply> <cn type='integer'> -322 </cn> </apply> </apply> <cn type='integer'> 28 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <eq /> <ci> w </ci> <apply> <ci> InverseGammaRegularized </ci> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

InverseGammaRegularized[a,z₁,z₂]