Incomplete gamma function

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In mathematics, the gamma function is defined by a definite integral. The incomplete gamma function is defined as an integral function of the same integrand. There are two varieties of the incomplete gamma function: the upper incomplete gamma function is for the case that the lower limit of integration is variable (ie where the "upper" limit is fixed), and the lower incomplete gamma function can vary the upper limit of integration.

The upper incomplete gamma function is defined as:

$\Gamma(a,x) = \int_x^{\infty} t^{a-1}\,e^{-t}\,dt .\,\!$

The lower incomplete gamma function is defined as:

$\gamma(a,x) = \int_0^x t^{a-1}\,e^{-t}\,dt .\,\!$

1 Properties
2 Regularized Gamma functions
3 References
4 Miscellaneous utilities

[edit] Properties

In both cases, x is a real variable, with x greater than or equal to zero, and a is a complex variable, such that the real part of a is positive.

By integration by parts we can find that

$\Gamma(a+1,x) = a\Gamma(a,x) + x^a e^{-x}\,$

$\gamma(a+1,x) = a\gamma(a,x) - x^a e^{-x}.\,$

Since the ordinary gamma function is defined as

$\Gamma(a) = \int_0^{\infty} t^{a-1}\,e^{-t}\,dt \,\!$

we have

$\gamma(a,x) + \Gamma(a,x) = \Gamma(a).\,$

Furthermore,

$\Gamma(a,x)=(a-1)!e^{-x}\sum_{k=0}^{a-1}\frac{x^k}{k!}$ if a is an integer. (Weisstein)

$\Gamma(a,0) = \Gamma(a)\,$

$\Gamma(a) = (a-1)!\,$ if a is an integer.

and

$\gamma(a,x) \rightarrow \Gamma(a) \quad \mathrm{as\ } x \rightarrow \infty. \,$

Also

$\Gamma(0,x) = -\mbox{Ei}(-x)\mbox{ for }x>0 \,$

$\Gamma\left({1 \over 2}, x\right) = \sqrt\pi\,\mbox{erfc}\left(\sqrt x\right) \,$

$\gamma\left({1 \over 2}, x\right) = \sqrt\pi\,\mbox{erf}\left(\sqrt x\right) \,$

$\Gamma(1,x) = e^{-x} \,$

$\gamma(1,x) = 1 - e^{-x} \,$

where Ei is the exponential integral, erf is the error function, and erfc is the complementary error function, erfc(x) = 1 − erf(x).

[edit] Regularized Gamma functions

Two related functions are the regularized Gamma functions:

$P(a,x)=\frac{\gamma(a,x)}{\Gamma(a)}$

$Q(a,x)=\frac{\Gamma(a,x)}{\Gamma(a)}=1-P(a,x)$

[edit] References

M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See section §6.5.)

G. Arfken and H. Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. (See Chapter 10.)

Armido R. DiDonato, Alfred H. Morris, Jr., Computation of the incomplete gamma function ratios and their inverse, ACM Transactions on Mathematical Software (TOMS), v.12 n.4, p.377-393, Dec. 1986.

Armido R. DiDonato, Alfred H. Morris, Jr., ALGORITHM 654: FORTRAN subroutines for computing the incomplete gamma function ratios and their inverse, ACM Transactions on Mathematical Software (TOMS), v.13 n.3, p.318-319, Sept. 1987. (See also Fortran 90 subroutines for ALGORITHM 654.)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Section 6.2.)