# gamma function

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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) Incomplete GammaFunctions
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)). …
##### 5: 8.2 Definitions and Basic Properties
###### §8.2 Definitions and Basic Properties
The general values of the incomplete gamma functions $\gamma\left(a,z\right)$ and $\Gamma\left(a,z\right)$ are defined by …
###### §8.2(ii) Analytic Continuation
In this subsection the functions $\gamma$ and $\Gamma$ have their general values. …
##### 6: 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: 5.1 Special Notation
 $j,m,n$ nonnegative integers. …
The main functions treated in this chapter are the gamma function $\Gamma\left(z\right)$, the psi function (or digamma function) $\psi\left(z\right)$, the beta function $\mathrm{B}\left(a,b\right)$, and the $q$-gamma function $\Gamma_{q}\left(z\right)$. … Alternative notations for this function are: $\Pi(z-1)$ (Gauss) and $(z-1)!$. …
##### 10: 8.7 Series Expansions
###### §8.7 Series Expansions
8.7.2 $\gamma\left(a,x+y\right)-\gamma\left(a,x\right)=\Gamma\left(a,x\right)-\Gamma% \left(a,x+y\right)=e^{-x}x^{a-1}\sum_{n=0}^{\infty}\frac{{\left(1-a\right)_{n}% }}{(-x)^{n}}(1-e^{-y}e_{n}(y)),$ $|y|<|x|$.
8.7.3 $\Gamma\left(a,z\right)=\Gamma\left(a\right)-\sum_{k=0}^{\infty}\frac{(-1)^{k}z% ^{a+k}}{k!(a+k)}=\Gamma\left(a\right)\left(1-z^{a}e^{-z}\sum_{k=0}^{\infty}% \frac{z^{k}}{\Gamma\left(a+k+1\right)}\right),$ $a\neq 0,-1,-2,\dots$.
8.7.4 $\gamma\left(a,x\right)=\Gamma\left(a\right)x^{\frac{1}{2}a}e^{-x}\sum_{n=0}^{% \infty}e_{n}(-1)x^{\frac{1}{2}n}I_{n+a}\left(\textstyle 2x^{1/2}\right),$ $a\neq 0,-1,-2,\dots$.
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).