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Incomplete Gamma Function

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The "complete" gamma function Gamma(a) can be generalized to the incomplete gamma function Gamma(a,x) such that Gamma(a)=Gamma(a,0). This "upper" incomplete gamma function is given by

Gamma(a,x)=int_x^inftyt^(a-1)e^(-t)dt.
(1)

For a an integer n

Gamma(n,x)=(n-1)!e^(-x)sum_(k=0)^(n-1)(x^k)/(k!)
(2)
=(n-1)!e^(-x)e_(n-1)(x),
(3)

where e_n(x) is the exponential sum function. It is implemented as Gamma[a, z] in the Wolfram Language.

The incomplete gamma function Gamma(0,x) has continued fraction

Gamma(0,x)=(e^(-x))/(x+1-1/(x+3-4/(x+5-9/(x+7+...))))
(4)

(Wall 1948, p. 358).

The lower incomplete gamma function is given by

gamma(a,x)=int_0^xt^(a-1)e^(-t)dt
(5)
=a^(-1)x^ae^(-x)_1F_1(1;1+a;x)
(6)
=a^(-1)x^a_1F_1(a;1+a;-x),
(7)

where _1F_1(a;b;x) is the confluent hypergeometric function of the first kind. For a an integer n,

gamma(n,x)=(n-1)!(1-e^(-x)sum_(k=0)^(n-1)(x^k)/(k!))
(8)
=(n-1)![1-e^(-x)e_(n-1)(x)].
(9)

It is implemented as Gamma[a, 0, z] in the Wolfram Language.

By definition, the lower and upper incomplete gamma functions satisfy

Gamma(a,x)+gamma(a,x)=Gamma(a).
(10)

The exponential integral Ei(z) is closely related to the incomplete gamma function Gamma(0,z) by

Gamma(0,z)=-Ei(-z)+1/2[ln(-z)-ln(-1/z)]-lnz.
(11)

Therefore, for real x,

Gamma(0,x)={-Ei(-x)-ipi for x<0; -Ei(-x) for x>0.
(12)

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