Inverse of the regularized incomplete gamma function: Differentiation (formula 06.12.20.0012)

InverseGammaRegularized

$Mathematica Notation$

Gamma, Beta, Erf

InverseGammaRegularized[a,z]

Differentiation

Low-order differentiation

With respect to z

http://functions.wolfram.com/06.12.20.0012.01

Input Form

D[InverseGammaRegularized[a, z], {z, 10}] == E^(10 w) w^(1 - 10 a) (1 - a (45 + a (-870 + a (9450 + a (-63273 + a (269325 + 4 a (-180920 + 9 a (32575 + 4 a (-7129 + 2520 a)))))))) + 1013 w + a (-23558 + a (236708 + a (-1342358 + a (4695827 + 4 a (-2587913 + 9 a (388057 + 28 a (-10369 + 3240 a))))))) w - 12 (-1 + a) (-1 + 2 a) (-1 + 3 a) (5132 + a (-48309 + a (173515 + 42 a (-6769 + 4320 a)))) w^2 + 10 (-1 + a) (-1 + 2 a) (92035 + a (-726451 + 36 a (60397 + 7 a (-11665 + 6048 a)))) w^3 - 4 (-1 + a) (1297853 + a (-8822594 + 45 a (503655 + 28 a (-20737 + 9072 a)))) w^4 + 12 (-1 + a) (-1110757 + 63 a (79109 + 4 a (-29809 + 15120 a))) w^5 - 72 (-1 + a) (235275 + 14 a (-44929 + 30240 a)) w^6 + 144 (-1 + a) (-75169 + 90720 a) w^7 - 3265920 (-1 + a) w^8 + 362880 w^9) Gamma[a]^10 /; w == InverseGammaRegularized[a, z]

Standard Form

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MathML Form

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</mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6048 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 11665 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 60397 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 726451 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 92035 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> w </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <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> 42 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4320 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 6769 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 173515 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 48309 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 5132 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> w </mi> <mn> 2 </mn> </msup> </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> <mn> 9 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 28 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ 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⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <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> <mn> 2520 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 7129 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 32575 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 180920 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 269325 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 63273 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 9450 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 870 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 45 </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> 10 </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'> 10 </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'> 10 </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'> 10 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <plus /> 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Date Added to functions.wolfram.com (modification date)

2007-05-02

InverseGammaRegularized[a,z₁,z₂]