Incomplete gamma function: Definition from Answers.com

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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:

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

The lower incomplete gamma function is defined as:

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

1 Properties
2 Evaluation Formulae
3 Regularized Gamma functions and Poisson random variables
4 Derivatives
5 Notes
6 References
7 Miscellaneous utilities

Properties

In both cases s is a complex parameter, such that the real part of s is positive.

By integration by parts (or $Δ$ times repeated) we find

$\begin{align} \Gamma(s,x)& = (s-1)\Gamma(s-1,x) + x^{s-1} e^{-x}\\ &={s-1 \choose \Delta} \Delta! \Gamma(s-\Delta,x)+ x^s e^{-x} \cdot \sum_{i=0}^{\Delta-1} {s-1 \choose i} \frac{i!}{x^{i+1}} \\ &={s-1 \choose \Delta} \Delta! \Gamma(s-\Delta,x)+ x^{s-\Delta} e^{-x} \cdot \sum_{i=0}^{\Delta-1} (-1)^i L_i^{(s-\Delta)}(x) \frac{\Delta! {s-1 \choose \Delta-1-i}}{i+1} \\ &=\frac{\Gamma(s+1,x)-x^s e^{-x}}{s}\\ &=\frac{\Gamma(s+\Delta,x)}{\Delta! {s-1+\Delta \choose \Delta}}-x^s e^{-x} \cdot \sum_{i=0}^{\Delta-1} \frac{x^i}{(i+1)! {s+i \choose i+1}}\\ &=\frac{\Gamma(s+\Delta,x)}{\Delta! {s-1+\Delta \choose \Delta}}-x^s e^{-x} \cdot \sum_{i=0}^{\Delta-1} (-1)^i\frac{{\Delta \choose i+1} L_i^{(s-1-i)}(x)}{(i+1) {\Delta+s-1 \choose i+1}}, \end{align}$

and conversely

$\begin{align} \gamma(s,x) &=(s-1)\gamma(s-1,x) - x^{s-1} e^{-x}\\ &={s-1 \choose \Delta} \Delta! \gamma(s-\Delta,x)- x^s e^{-x} \cdot \sum_{i=0}^{\Delta-1} {s-1 \choose i} \frac{i!}{x^{i+1}} \\ &={s-1 \choose \Delta} \Delta! \gamma(s-\Delta,x)- x^{s-\Delta} e^{-x} \cdot \sum_{i=0}^{\Delta-1} (-1)^i L_i^{(s-\Delta)}(x) \frac{\Delta! {s-1 \choose \Delta-1-i}}{i+1} \\ &=\frac{\gamma(s+1,x)+x^s e^{-x}}{s}\\ &=x^s e^{-x} \sum_{i=0}^{\Delta-1} \frac{x^i}{(i+1)! {s+i \choose i+1}}+\frac{\gamma(s+\Delta,x)}{\Delta! {s-1+\Delta \choose \Delta}}\\ &=x^s e^{-x} \sum_{i=0}^{\Delta-1} (-1)^i\frac{L_i^{(s-1-i)}(x){\Delta \choose i+1}}{(i+1){s-1+\Delta\choose i+1}}+\frac{\gamma(s+\Delta,x)}{\Delta! {s-1+\Delta \choose \Delta}}\\ &=x^s e^{-x} \sum_{i=0}^{\Delta-1} (-1)^i\frac{1+i}{s+i}\frac{x^i}{(i+1)!}\sum_{j=0}^{\Delta-1-i}\frac{x^j}{j!}+\frac{\gamma(s+\Delta,x)}{\Delta! {s-1+\Delta \choose \Delta}},\end{align}$

where $L$ is Laguerre's polynomial.

Since the ordinary gamma function is defined as

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

we have

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

Some selected properties of incomplete gamma function:

$\Gamma(s) = (s-1)!\,$ if s is a positive integer;

$\Gamma(s,x)=(s-1)!e^{-x}\sum_{k=0}^{s-1}\frac{x^k}{k!}$ if s is a positive integer;^[1]

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

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

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

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

$\Gamma(s,x) = x^s \cdot \mbox{Ei}_{1-s}(x), \,$

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

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

$\frac{\gamma(s,x)}{x^s} \rightarrow \frac 1 s \quad \mathrm{as\ } x \rightarrow 0, \,$

$\frac{\Gamma(s,x)}{x^s} \rightarrow -\frac 1 s \quad \mathrm{as\ } x \rightarrow 0$ and $\Re (s) < 0\,$

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

$\frac{\Gamma(s,x)}{x^{s-1} e^{-x}} \rightarrow 1 \quad \mathrm{as\ } x \rightarrow \infty, \,$

$\int_0^\infty \frac{x^{s-1}e^{-x}}{\Gamma(s)} \frac{\Gamma(s,x)}{\Gamma(s)}dx= \frac{1}{2},$

$\frac{1}{\Gamma(s)} \int_z^\infty \Psi_{s,a}(x,z) x^{s-1}e^{-x} dx = \frac{1}{\Gamma(a)} \int_z^\infty x^{a-1}e^{-x}dx , \mathrm{with\ }\$

$\Psi_{s,a}(x,z)=1-I_{z/x}(a,s-a), s > a > 0, z > 0 , \mathrm{and\ } I_{z/x}(a,s-a)$ is the regularized incomplete beta function^[2]

$\int_x^\infty t^{b-1} \Gamma(s,t)dt= \frac{1}{b}\left(\Gamma(s+b,x)- x^b \Gamma(s,x) \right),$ and $\int_0^x t^{b-1} \Gamma(s,t)dt= \frac{1}{b}\left(\gamma(s+b,x)+ x^b \Gamma(s,x) \right),$

$\Gamma(s,x) \sim x^{s-1} e^{-x} \cdot \sum_{i=0} \frac{(s-1)!}{(s-i-1)!}x^{-i}$ as an asymptotic series as s → ∞ with |arg s| < 3π/2.

$\frac{\Gamma(s+n,x)}{n! {s+n-1 \choose n}}\rightarrow \Gamma(s)$ , as $n \to \infty.$

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

Evaluation Formulae

The lower gamma function has the straight forward expansion

$\gamma(s,z)= \sum_{k=0}^\infty \frac{(-1)^k}{k!} \frac{z^{s+k}}{s+k}= \frac{z^s}{s} M(s, s+1,-z),$

where M is Kummer's confluent hypergeometric function.

Another form of a series expansion is given by

$\gamma(a,x)= x^a e^{-x} \sum_{n=0}^\infty \frac{x^n}{a(a+1)...(a+n)}$

(Ref: http://algolist.manual.ru/maths/count_fast/gamma_function.php ^[3] )

This has shown greater accuracy in computing the Gamma distribution CDF, perhaps because it is not an alternating series.

Connection with Kummer's confluent hypergeometric function

It is easily shown that, when the real part of z is positive,

$\gamma(s,z) = \int_0^z e^{-t}t^{s-1} dt = s^{-1} z^s e^{-z} M(1,s+1,z)= z^s e^{-z} \sum_{k=0} \frac {z^k}{(k+1)! {s+k \choose k+1}}.$

Since the series

$M(1, s+1, z) = 1 + \frac{z}{(s+1)}+ \frac{z^2}{(s+1)(s+2)}+ \frac{z^3}{(s+1)(s+2)(s+3)}+ \cdots$

has an infinite radius of convergence, we may take

$\gamma(s,z) = s^{-1} z^s e^{-z} M(1,s+1,z)\,$

as the definition of γ(s, z) for all complex z. In this light, the lower incomplete gamma function γ(s, z) is an entire function of the complex variable z. Since the gamma function Γ(z) is a meromorphic function with simple poles at {0, −1, −2, …}, we may define the meromorphic upper incomplete gamma function as

$\Gamma(s, z) = \Gamma(s) - \gamma(s, z). \,$

Again with confluent hypergeometric functions and employing Kummer's identity,

$\begin{align}\Gamma(s,z)=& e^{-z} U(1-s,1-s,z)=\frac{z^s e^{-z}}{\Gamma(1-s)} \int_0^\infty \frac{e^{-u}}{u^s (z+u)}\mathrm{d}u=\\ =& e^{-z} z^s U(1,1+s,z)= e^{-z} \int_0^\infty e^{-u} (z+u)^{s-1}\mathrm{d} u= e^{-z} z^s \int_0^\infty e^{-z u} (1+u)^{s-1}\mathrm{d} u.\end{align}$

For the actual computation of numerical values, Gauss's continued fraction provides a useful expansion:

$\gamma(s, z) = \cfrac{z^s e^{-z}}{s - \cfrac{s z}{s+1 + \cfrac{z}{s+2 - \cfrac{(s+1)z} {s+3 + \cfrac{2z}{s+4 - \cfrac{(s+2)z}{s+5 + \cfrac{3z}{s+6 - \ddots}}}}}}}.$

This continued fraction converges for all complex z, provided only that s is not a negative integer.

The upper gamma function has the continued fraction

$\Gamma(s, z) = \cfrac{z^s e^{-z}}{z+\cfrac{1-s}{1 + \cfrac{1}{z + \cfrac{2-s} {1 + \cfrac{2}{z+ \cfrac{3-s}{1+ \ddots}}}}}}$ ^[4]

and

$\Gamma(s, z)= \cfrac{z^s e^{-z}}{1+z-s+ \cfrac{s-1}{3+z-s+ \cfrac{2(s-2)}{5+z-s+ \cfrac{3(s-3)} {7+z-s+ \cfrac{4(s-4)}{9+s-z+ \ddots}}}}}.$

Connection with Laguerre polynomials

The incomplete Gamma functions have a series representation in terms of Laguerre polynomials, as

$\begin{align}\gamma(s;z)=& \left(\frac{z}{1+t}\right)^s \cdot \sum_{k=0}^\infty \frac{L_k^{(s)}\left(\frac{z}{t} \right)}{s+k} \left( \frac{t}{1+t} \right)^k \\ =& z^s e^{-z} \cdot \sum_{k=0}(-1)^k\frac{L_k^{(s-1-k)}(z)}{k+1} \\ =& -z^s e^{-z} \cdot \sum_{k=0}\frac{L_k^{(\alpha-k)}(z)}{(k+1) {\alpha-s \choose k+1}};\end{align}$

if $α$ is a positive integer this reduces further to

$\frac{\gamma(s;z)}{z^s e^{-z}}= \alpha!\cdot \sum_{k=0} \frac{(-1)^{k+1-\alpha}}{(k+1)!} \frac{z^{k-\alpha} L_\alpha^{(k-\alpha)}(z)}{{\alpha-s \choose k+1}}.$

Moreover,

$\begin{align} \Gamma(s;z)&=\left( \frac{z}{t} \right)^s \cdot \sum_{k=0} \frac{1-\left(1-\frac{1}{t+1}\right)^{s+k}}{s+k} L_k^{(s)}\left( \frac{z}{t}\right) \\ &= z^s e^{-z} \cdot \sum_{k=0} \frac{L_k^{(s)}(z)}{k+1} \qquad \left(\Re(s)< \frac 1 2\right)\\ &=z^s e^{- z} \sum_{k=0} \frac{L_k^{(\alpha)}(z)}{(k+1) {k+1+\alpha-s \choose k+1}} \qquad \left(\Re\left(s-\frac \alpha 2 \right)< \frac 1 4 \right)\\ &= z^s e^{-z} \left(\sum_{k=0}^{\Delta-1} {s-1 \choose k} \frac{k!}{z^{k+1}} +{s-1 \choose \Delta} \frac{\Delta!}{z^\Delta} \cdot \sum_{k=0} \frac{L_k^{\left(-\frac 1 2-\Delta \right)}(z)}{(k+1) {\frac 1 2-s+k \choose k+1}}\right) \qquad (\Re (s)< \frac \Delta 2)\\ &= \left(\frac{z}{t}\right)^s e^{-\frac{z}{t} } \cdot \sum_{k=0} L_k^{(s)}\left(\frac{z}{t}\right) \frac{1-\left(1-\frac{1}{t}\right)^{k+1}}{k+1}\\ &= \left(\frac{z}{t}\right)^s e^{-\frac{z}{t}} \cdot \sum_{k=0} L_k^{(s+\Delta)}\left(\frac{z}{t}\right) \sum_{i=0}^{\Delta} (-1)^i {\Delta \choose i} \frac{1-\left(1-\frac{1}{t} \right)^{i+k+1}}{i+k+1}\\ &= \left(\frac{z}{t}\right)^s e^{-\frac{z}{t}} \cdot \sum_{k=0} \frac{L_k^{(s+\Delta)}\left(\frac{z}{t}\right)}{(\Delta+1) {k+\Delta+1\choose \Delta+1}} \left(1-\left(\frac{t-1}{t}\right)^{k+1} \sum_{j=0}^\Delta \frac{{k+j \choose j}}{t^j}\right)\\ &=\left(\frac{z}{t}\right)^s e^{-z} \cdot \sum_{k=0} \frac{L_k^{(s)}\left(\frac{z}{t}\right)}{(k+1)(t-1)^{k+1}} \frac{t^{k+s}- \sum_{j=0}^k {s+k \choose j} (t-1)^j}{{s+k \choose k+1}}\\ &= \frac{(\alpha+s)!}{(\alpha+1)!} e^{-z} \sum_{k=0} L_k^{(\alpha)}(z)\frac {{s-1 \choose k}}{{-\alpha-2 \choose k}}.\end{align}$

Connection with Bessel functions

The lower incomplete gamma function has the following representations in terms of Bessel functions:

$\begin{align}\gamma(s,z)&=\frac{2 \sqrt \pi}{s} z^{s-\frac{1}{2}} e^{-\frac{z}{2}} \sum_{n=0} \frac{{s-1 \choose n}}{{-s-1 \choose n}} \left( n+\frac{1}{2} \right) I_{n+\frac{1}{2}}\left( \frac{z}{2} \right)\\ &= \frac{1}{2} 4^s \sqrt z e^{-\frac{z}{2}} \Gamma \left( s-\frac{1}{2}\right) \sum_{n=0} \frac{(s-1){1-2s \choose n} }{(s+n-1)(s+n)} \left(s+n-\frac{1}{2}\right) I_{s+n-\frac{1}{2}} \left(\frac{z}{2} \right)\end{align}$

Connection with Fourier Transform

The lower and the upper incomplete Gamma function are each other's Fourier transform, as

$\int_{-\infty}^\infty \frac{\gamma\left(\frac s 2, z^2 \pi \right)}{(z^2 \pi)^\frac s 2} e^{-2 \pi i k z}\mathrm d z= \frac{\Gamma\left(\frac {1-s} 2, k^2 \pi \right)}{(k^2 \pi)^\frac {1-s} 2};$

consequently, by Poisson's summation formula,

$\sum_{k=1}\frac{\Gamma\left(\frac {1-s} 2, \frac{k^2 \pi} x\right)}{(\frac{k^2 \pi} x)^{\frac {1-s} 2}}= \frac \sqrt x s+ \frac 1 {1-s}+ \sqrt x \cdot \sum_{k=1}\frac{\gamma\left(\frac s 2, k^2 \pi x\right)}{(k^2 \pi x)^{\frac s 2}}.$

Multiplication theorem

The following multiplication theorem holds true:

$\begin{align}\Gamma(s,z)&=\frac 1 {t^s} \sum_{i=0} \frac{\left(1-\frac 1 t \right)^i}{i!} \Gamma(s+i,t z)\\ &= \Gamma(s,t z) -(t z)^s e^{-t z} \sum_{i=1} \frac{\left(\frac 1 t-1 \right)^i}{i} L_{i-1}^{(s-i)}(t z).\end{align}$

Regularized Gamma functions and Poisson random variables

Two related functions are the regularized Gamma functions:

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

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

When $s\geq 0$ is an integer, $Q (s,λ)$ is the cumulative distribution function for Poisson random variables: If $X$ is a $P o i (λ)$ random variable then

$Pr(X<s) = \sum_{i<s} e^{-\lambda} \frac{\lambda^i}{i!} = \frac{\Gamma(s,\lambda)}{\Gamma(s)} = Q(s,\lambda).$

This formula can be derived by repeated integration by parts.

Derivatives

The derivative of the upper incomplete gamma function $Γ(s, x)$ with respect to x is well known. It is simply given by the integrand of its integral definition:

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

The derivative with respect to its first argument "s" is given by ^[5]

$\frac{\partial \Gamma (s,x) }{\partial s} = \ln (x) \Gamma (s,x) + x ~T(3,s,x)$

and the second derivative is:

$\frac{\partial^2 \Gamma (s,x) }{\partial s^2} = \ln^2 (x) \Gamma (s,x) + 2 x ~ ( \ln (x) ~ T(3,s,x) + T(4,s,x) )$

where the function "T(m,s,x)" is a special case of the Meijer G-function

$T(m,s,z) = G_{m-1, m}^{~m,~0} \left( x \left| \begin{array}{c} 0,0, \ldots 0 \\ -1, -1, \ldots, s-1, -1 \end{array} \right. \right) ~.$

This particular special case has internal closure properties of its own because it can be used to express all successive derivatives. In general,

$\frac{\partial^m \Gamma (s,x) }{\partial s^m} = \ln^m (x) \Gamma (s,x) + m x ~ \sum_{i=0}^{m-1} P_i^{m-1} \ln^{m-i-1} (x) ~ T(3+i,s,x)$

where

$P_j^i = \left( \begin{array}{l} i \\ j \end{array} \right) j! = \frac{i!}{(i-j)!} ~ .$

All such derivatives can be generated in succession from:

$\frac{\partial T (m,s,x) }{\partial s} = \ln (x) ~ T(m,s,x) + (m-1) T(m+1,s,x)$

and

$\frac{\partial T (m,s,x) }{\partial x} = -\frac{1}{x} (T(m-1,s,x) + T(m,s,x))$

This function T(m,s,x) can be computed from its series representation valid for : $| z | < 1$ ,

$T(m,s,z) = - \frac{(-1)^{m-1} }{(m-2)! } \frac{d^{m-2} }{dt^{m-2} } \left. (\Gamma (s-t) z^{t-1} ) \right]_{t=0} + \sum_{i=0}^{\infty} \frac{(-1)^i z^{s-1+i}}{i! (-s-i)^{m-1} }$

with the understanding that s is not a negative integer or zero. In such a case, one must use a limit. Results for $|z| \ge 1$ can be obtained by analytic continuation. Some special cases of this function can be simplified. For example,

$T(2,s,x) = \frac{\Gamma(s,x)}{x}$

$x ~ T(3,1,x) = E_1 (x)$

where $E 1 (x)$ is the Exponential integral. These derivatives and the function T(m,s,x) provides exact solutions to a number of integrals by repeated differentiation of the integral definition of the upper incomplete gamma function. For example,

$\int_{x}^{\infty} t^{s-1} \ln^m (t) ~ e^{-t} = \frac{\partial^m}{\partial s^m} \int_{x}^{\infty} t^{s-1} e^{-t} = \frac{\partial^m}{\partial s^m} \Gamma (s,x)$

This formula can be further inflated or generalized to a huge class of Laplace transforms and Mellin transforms. When combined with a computer algebra system, the exploitation of special functions provides a powerful method for solving definite integrals, in particular those encountered by practical engineering applications.

Notes

^ Weisstein, Eric W., "Incomplete Gamma Function" from MathWorld. (equation 2)
^ C. Canepa J. Phys. Chem. A 2006, 110, 13290-13294; http://sites.google.com/site/carlocanepa8/home/math/
^ http://algolist.manual.ru/maths/count_fast/gamma_function.php
^ Abramowitz and Stegunp. 263, 6.5.31
^ K.O Geddes, M.L. Glasser, R.A. Moore and T.C. Scott, Evaluation of Classes of Definite Integrals Involving Elementary Functions via Differentiation of Special Functions, AAECC (Applicable Algebra in Engineering, Communication and Computing), vol. 1, (1990), pp.149-165, [1]

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.)

Miscellaneous utilities

$P (a, x)$ - Incomplete Gamma Function Calculator
$Q (a, x)$ - Incomplete Gamma Function Calculator - Complemented
$γ(a, x)$ - Incomplete Gamma Function Calculator - Lower Limit of Integration
$Γ(a, x)$ - Incomplete Gamma Function Calculator - Upper Limit of Integration
formulas and identities of the Incomplete Gamma Function functions.wolfram.com

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