Inverse of the regularized incomplete gamma function: Differentiation (formula 06.12.20.0011)

InverseGammaRegularized

$Mathematica Notation$

Gamma, Beta, Erf

InverseGammaRegularized[a,z]

Differentiation

Low-order differentiation

With respect to z

http://functions.wolfram.com/06.12.20.0011.01

Input Form

D[InverseGammaRegularized[a, z], {z, 9}] == (-E^(9 w)) w^(1 - 9 a) (1 + a (-36 + a (546 + a (-4536 + a (22449 + 4 a (-16821 + a (29531 + 36 a (-761 + 280 a))))))) + 502 w - 2 a (4679 + a (-36893 + a (159365 + 4 a (-101576 + a (152039 + 36 a (-3403 + 1120 a)))))) w + 2 (-1 + a) (-1 + 2 a) (9511 + a (-87099 + 4 a (75437 + 45 a (-2619 + 1568 a)))) w^2 - 8 (-1 + a) (-1 + 2 a) (-23411 + 4 a (31337 + 45 a (-1265 + 784 a))) w^3 + 4 (-1 + a) (-175291 + a (829183 + 180 a (-7323 + 3920 a))) w^4 - 8 (-1 + a) (146221 + 36 a (-11243 + 7840 a)) w^5 + 144 (-1 + a) (-6361 + 7840 a) w^6 - 322560 (-1 + a) w^7 + 40320 w^8) Gamma[a]^9 /; w == InverseGammaRegularized[a, z]

Standard Form

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MathML Form

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</mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 152039 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 101576 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 159365 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 36893 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 4679 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> w </mi> </mrow> <mo> + </mo> <mrow> <mn> 502 </mn> <mo> ⁢ </mo> <mi> w </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 36 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 280 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 761 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 29531 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 16821 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 22449 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 4536 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 546 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 36 </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> 9 </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'> 9 </cn> </degree> </bvar> <apply> <ci> InverseGammaRegularized </ci> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 9 </cn> <ci> w </ci> </apply> </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'> 9 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 40320 </cn> <apply> <power /> <ci> w </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 322560 </cn> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> w </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 144 </cn> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 7840 </cn> <ci> a </ci> </apply> <cn type='integer'> -6361 </cn> </apply> <apply> <power /> <ci> w </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 36 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 7840 </cn> <ci> a </ci> </apply> <cn type='integer'> -11243 </cn> </apply> </apply> <cn type='integer'> 146221 </cn> </apply> <apply> <power /> <ci> w </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 180 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3920 </cn> <ci> a </ci> </apply> <cn type='integer'> -7323 </cn> </apply> </apply> <cn type='integer'> 829183 </cn> </apply> </apply> <cn type='integer'> -175291 </cn> </apply> <apply> <power /> <ci> w </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </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 /> <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'> 784 </cn> <ci> a </ci> </apply> <cn type='integer'> -1265 </cn> </apply> </apply> <cn type='integer'> 31337 </cn> </apply> </apply> <cn type='integer'> -23411 </cn> </apply> <apply> <power /> <ci> w </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </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'> 4 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 45 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 1568 </cn> <ci> a </ci> </apply> <cn type='integer'> -2619 </cn> </apply> </apply> <cn type='integer'> 75437 </cn> </apply> </apply> <cn type='integer'> -87099 </cn> </apply> </apply> <cn type='integer'> 9511 </cn> </apply> <apply> <power /> <ci> w </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <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 /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 36 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 1120 </cn> <ci> a </ci> </apply> <cn type='integer'> -3403 </cn> </apply> </apply> <cn type='integer'> 152039 </cn> </apply> </apply> <cn type='integer'> -101576 </cn> </apply> </apply> <cn type='integer'> 159365 </cn> </apply> </apply> <cn type='integer'> -36893 </cn> </apply> </apply> <cn type='integer'> 4679 </cn> </apply> <ci> w </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 502 </cn> <ci> w </ci> </apply> <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 /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 36 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 280 </cn> <ci> a </ci> </apply> <cn type='integer'> -761 </cn> </apply> </apply> <cn type='integer'> 29531 </cn> </apply> </apply> <cn type='integer'> -16821 </cn> </apply> </apply> <cn type='integer'> 22449 </cn> </apply> </apply> <cn type='integer'> -4536 </cn> </apply> </apply> <cn type='integer'> 546 </cn> </apply> </apply> <cn type='integer'> -36 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <cn type='integer'> 9 </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]