# incomplete gamma functions

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##### 1: 8.23 Statistical Applications
###### §8.23 Statistical Applications
The functions $P\left(a,x\right)$ and $Q\left(a,x\right)$ are used extensively in statistics as the probability integrals of the gamma distribution; see Johnson et al. (1994, pp. 337–414). …In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of $Q\left(a,x\right)$; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319).
##### 2: 8.16 Generalizations
###### §8.16 Generalizations
For a generalization of the incomplete gamma function, including asymptotic approximations, see Chaudhry and Zubair (1994, 2001) and Chaudhry et al. (1996). …
##### 3: 8.24 Physical Applications
###### §8.24(i) IncompleteGammaFunctions
The function $\gamma\left(a,x\right)$ appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)). …
##### 4: 7.16 Generalized Error Functions
These functions can be expressed in terms of the incomplete gamma function $\gamma\left(a,z\right)$8.2(i)) by change of integration variable.
##### 5: 8.2 Definitions and Basic Properties
###### §8.2(ii) Analytic Continuation
In this subsection the functions $\gamma$ and $\Gamma$ have their general values. The function $\gamma^{*}\left(a,z\right)$ is entire in $z$ and $a$. …
##### 6: 6.11 Relations to Other Functions
###### IncompleteGammaFunction
6.11.1 $E_{1}\left(z\right)=\Gamma\left(0,z\right).$
##### 7: 8.22 Mathematical Applications
###### §8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function
The function $\Gamma\left(a,z\right)$, with $|\operatorname{ph}a|\leq\tfrac{1}{2}\pi$ and $\operatorname{ph}z=\tfrac{1}{2}\pi$, has an intimate connection with the Riemann zeta function $\zeta\left(s\right)$25.2(i)) on the critical line $\Re s=\tfrac{1}{2}$. See Paris and Cang (1997). …
##### 8: 8.7 Series Expansions
###### §8.7 Series Expansions
8.7.2 $\gamma\left(a,x+y\right)-\gamma\left(a,x\right)=\Gamma\left(a,x\right)-\Gamma% \left(a,x+y\right)=e^{-x}x^{a-1}\sum_{n=0}^{\infty}\frac{{\left(1-a\right)_{n}% }}{(-x)^{n}}(1-e^{-y}e_{n}(y)),$ $|y|<|x|$.
8.7.4 $\gamma\left(a,x\right)=\Gamma\left(a\right)x^{\frac{1}{2}a}e^{-x}\sum_{n=0}^{% \infty}e_{n}(-1)x^{\frac{1}{2}n}I_{n+a}\left(\textstyle 2x^{1/2}\right),$ $a\neq 0,-1,-2,\dots$.
8.7.6 $\Gamma\left(a,x\right)=x^{a}e^{-x}\sum_{n=0}^{\infty}\frac{L^{(a)}_{n}\left(x% \right)}{n+1},$ $x>0$.
For an expansion for $\gamma\left(a,ix\right)$ in series of Bessel functions $J_{n}\left(x\right)$ that converges rapidly when $a>0$ and $x$ ($\geq 0$) is small or moderate in magnitude see Barakat (1961).
##### 9: 8.3 Graphics
###### §8.3(i) Real Variables
Some monotonicity properties of $\gamma^{*}\left(a,x\right)$ and $\Gamma\left(a,x\right)$ in the four quadrants of the ($a,x$)-plane in Figure 8.3.6 are given in Erdélyi et al. (1953b, §9.6). Figure 8.3.7: x - a - γ * ⁡ ( a , x ) (= x - a ⁢ Q ⁡ ( a , x ) ), 0 ≤ x ≤ 4 , - 5 ≤ a ≤ 5 . Magnify 3D Help
###### §8.3(ii) Complex Argument Figure 8.3.16: γ * ⁡ ( 2.5 , x + i ⁢ y ) , - 3 ≤ x ≤ 3 , - 3 ≤ y ≤ 3 . Magnify 3D Help
##### 10: 8.27 Approximations
###### §8.27(i) IncompleteGammaFunctions
• DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}e^{x^{2}/2}\int_{x}^{\infty}e^{-t^{2}/2}t^{p}\mathrm{d}t$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^{2}-p)+\sqrt{(x^{2}-p)^{2}+8(x^{2}+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to\infty$.

• Luke (1969b, pp. 25, 40–41) gives Chebyshev-series expansions for $\Gamma\left(a,\omega z\right)$ (by specifying parameters) with $1\leq\omega<\infty$, and $\gamma\left(a,\omega z\right)$ with $0\leq\omega\leq 1$; see also Temme (1994b, §3).