# gamma function

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## 11—20 of 349 matching pages

##### 11: 8.7 Series Expansions
###### §8.7 Series Expansions
8.7.1 $\gamma^{*}\left(a,z\right)=e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma\left(a% +k+1\right)}=\frac{1}{\Gamma\left(a\right)}\sum_{k=0}^{\infty}\frac{(-z)^{k}}{% k!(a+k)}.$
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 $\gamma\left(a,x\right)=\Gamma\left(a\right)x^{\frac{1}{2}a}e^{-x}\sum_{n=0}^{% \infty}e_{n}(-1)x^{\frac{1}{2}n}I_{n+a}\left(\textstyle 2x^{1/2}\right),$ $a\neq 0,-1,-2,\dots$.
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).
##### 12: 5.2 Definitions
###### Euler’s Integral
5.2.1 $\Gamma\left(z\right)=\int_{0}^{\infty}e^{-t}t^{z-1}\mathrm{d}t,$ $\Re z>0$.
5.2.2 $\psi\left(z\right)=\Gamma'\left(z\right)/\Gamma\left(z\right),$ $z\neq 0,-1,-2,\dots$.
5.2.5 ${\left(a\right)_{n}}=\Gamma\left(a+n\right)/\Gamma\left(a\right),$ $a\neq 0,-1,-2,\dots$.
##### 13: 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.
##### 14: 6.11 Relations to Other Functions
###### Incomplete GammaFunction
6.11.1 $E_{1}\left(z\right)=\Gamma\left(0,z\right).$
##### 15: 5.17 Barnes’ $G$-Function (Double Gamma Function)
###### §5.17 Barnes’ $G$-Function (Double GammaFunction)
5.17.3 $G\left(z+1\right)=(2\pi)^{z/2}\exp\left(-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\gamma z% ^{2}\right)\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp\left% (-z+\frac{z^{2}}{2k}\right)\right).$
5.17.4 $\operatorname{Ln}G\left(z+1\right)=\tfrac{1}{2}z\ln\left(2\pi\right)-\tfrac{1}% {2}z(z+1)+z\operatorname{Ln}\Gamma\left(z+1\right)-\int_{0}^{z}\operatorname{% Ln}\Gamma\left(t+1\right)\mathrm{d}t.$
5.17.5 $\operatorname{Ln}G\left(z+1\right)\sim\tfrac{1}{4}z^{2}+z\operatorname{Ln}% \Gamma\left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac{1}{12}\right)% \operatorname{Ln}z-\ln A+\sum_{k=1}^{\infty}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2% k}}.$
##### 16: 5.10 Continued Fractions
###### §5.10 Continued Fractions
5.10.1 $\operatorname{Ln}\Gamma\left(z\right)+z-\left(z-\tfrac{1}{2}\right)\ln z-% \tfrac{1}{2}\ln\left(2\pi\right)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}% {z+\cfrac{a_{3}}{z+\cfrac{a_{4}}{z+\cfrac{a_{5}}{z+}}}}}}\cdots,$
##### 17: 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
##### 19: 8.8 Recurrence Relations and Derivatives
###### §8.8 Recurrence Relations and Derivatives
If $w(a,z)=\gamma\left(a,z\right)$ or $\Gamma\left(a,z\right)$, then …
8.8.12 $Q\left(a+n,z\right)=Q\left(a,z\right)+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{% \Gamma\left(a+k+1\right)}.$
##### 20: 5.21 Methods of Computation
###### §5.21 Methods of Computation
Similarly for $\ln\Gamma\left(z\right)$, $\psi\left(z\right)$, and the polygamma functions. … For the computation of the $q$-gamma and $q$-beta functions see Gabutti and Allasia (2008).