Incomplete gamma function
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In mathematics, the gamma function is defined by a definite integral. The incomplete gamma function is defined by an indefinite integral of the same integrand.
There are two varieties of the incomplete gamma function, one for the case that the lower limit of integration is variable, and one for the upper limit of integration. The first is denoted <math>\Gamma(a,x)<math> and defined as
- <math> \Gamma(a,x) = \int_x^{\infty} t^{a-1}\,e^{-t}\,dt .\,\!<math>
The second is denoted <math>\gamma(a,x)<math> and defined as
- <math> \gamma(a,x) = \int_0^x t^{a-1}\,e^{-t}\,dt .\,\!<math>
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
- <math> \Gamma(a+1,x) = a\Gamma(a,x) + x^a e^{-x}\,<math>
- <math> \gamma(a+1,x) = a\gamma(a,x) - x^a e^{-x}.\,<math>
Since the ordinary gamma function is defined as
- <math> \Gamma(a) = \int_0^{\infty} t^{a-1}\,e^{-t}\,dt \,\!<math>
we have
- <math> \gamma(a,x) + \Gamma(a,x) = \Gamma(a).\, <math>
Furthermore,
- <math> \Gamma(a,0) = \Gamma(a)\, <math>
and
- <math> \gamma(a,x) \rightarrow \Gamma(a)
\quad \mathrm{as\ } x \rightarrow \infty. \,<math>
Also
- <math>\Gamma(0,x) = -\mbox{Ei}(-x)\mbox{ for }x>0 \,<math>
- <math>\Gamma\left({1 \over 2}, x\right) = \sqrt\pi\,\mbox{erfc}\left(\sqrt x\right) \,<math>
- <math>\gamma\left({1 \over 2}, x\right) = \sqrt\pi\,\mbox{erf}\left(\sqrt x\right) \,<math>
- <math>\Gamma(1,x) = e^{-x} \,<math>
- <math>\gamma(1,x) = 1 - e^{-x} \,<math>
where Ei is the exponential integral, erf is the error function, and erfc is the complementary error function, erfc(x) = 1 − erf(x).
Regularized Gamma functions
Two related functions are the regularized Gamma functions:
- <math>P(a,x)=\frac{\gamma(a,x)}{\Gamma(a)}<math>
- <math>Q(a,x)=\frac{\Gamma(a,x)}{\Gamma(a)}=1-P(a,x)<math>
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 (http://www.math.sfu.ca/~cbm/aands/page_260.htm).)
- G. Arfken and H. Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. (See Chapter 10.)
- 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.)ja:不完全ガンマ関数