# §8.4 Special Values

For $\operatorname{erf}\left(z\right)$, $\operatorname{erfc}\left(z\right)$, and $F\left(z\right)$, see §§7.2(i), 7.2(ii). For $E_{n}\left(z\right)$ see §8.19(i).

 8.4.1 $\gamma\left(\tfrac{1}{2},z^{2}\right)=2\int_{0}^{z}e^{-t^{2}}\mathrm{d}t=\sqrt% {\pi}\operatorname{erf}\left(z\right),$
 8.4.2 $\displaystyle\gamma^{*}\left(a,0\right)$ $\displaystyle=\frac{1}{\Gamma\left(a+1\right)},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function and $a$: parameter Permalink: http://dlmf.nist.gov/8.4.E2 Encodings: TeX, pMML, png See also: Annotations for 8.4 and 8 8.4.3 $\displaystyle\gamma^{*}\left(\tfrac{1}{2},-z^{2}\right)$ $\displaystyle=\frac{2e^{z^{2}}}{z\sqrt{\pi}}F\left(z\right).$
 8.4.4 $\Gamma\left(0,z\right)=\int_{z}^{\infty}t^{-1}e^{-t}\mathrm{d}t=E_{1}\left(z% \right),$
 8.4.5 $\Gamma\left(1,z\right)=e^{-z},$ ⓘ Symbols: $\mathrm{e}$: base of exponential function, $\Gamma\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function and $z$: complex variable Permalink: http://dlmf.nist.gov/8.4.E5 Encodings: TeX, pMML, png See also: Annotations for 8.4 and 8
 8.4.6 $\Gamma\left(\tfrac{1}{2},z^{2}\right)=2\int_{z}^{\infty}e^{-t^{2}}\mathrm{d}t=% \sqrt{\pi}\operatorname{erfc}\left(z\right).$

For $n=0,1,2,\dots$,

 8.4.7 $\displaystyle\gamma\left(n+1,z\right)$ $\displaystyle=n!(1-e^{-z}e_{n}(z)),$ 8.4.8 $\displaystyle\Gamma\left(n+1,z\right)$ $\displaystyle=n!e^{-z}e_{n}(z),$ 8.4.9 $\displaystyle P\left(n+1,z\right)$ $\displaystyle=1-e^{-z}e_{n}(z),$ ⓘ Symbols: $\mathrm{e}$: base of exponential function, $P\left(\NVar{a},\NVar{z}\right)$: normalized incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_{n}(z)$: functions A&S Ref: 6.5.13 Permalink: http://dlmf.nist.gov/8.4.E9 Encodings: TeX, pMML, png See also: Annotations for 8.4 and 8 8.4.10 $\displaystyle Q\left(n+1,z\right)$ $\displaystyle=e^{-z}e_{n}(z),$

where

 8.4.11 $e_{n}(z)=\sum_{k=0}^{n}\frac{z^{k}}{k!}.$ ⓘ Defines: $e_{n}(z)$: functions (locally) Symbols: $!$: factorial (as in $n!$), $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer A&S Ref: 6.5.11 Referenced by: §8.11(v), §8.7 Permalink: http://dlmf.nist.gov/8.4.E11 Encodings: TeX, pMML, png See also: Annotations for 8.4 and 8

Also

 8.4.12 $\gamma^{*}\left(-n,z\right)=z^{n},$ ⓘ Symbols: $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $z$: complex variable and $n$: nonnegative integer A&S Ref: 6.5.14 Referenced by: §8.13(i) Permalink: http://dlmf.nist.gov/8.4.E12 Encodings: TeX, pMML, png See also: Annotations for 8.4 and 8
 8.4.13 $\Gamma\left(1-n,z\right)=z^{1-n}E_{n}\left(z\right),$ ⓘ Symbols: $E_{\NVar{p}}\left(\NVar{z}\right)$: generalized exponential integral, $\Gamma\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $z$: complex variable and $n$: nonnegative integer Referenced by: §8.19(iv) Permalink: http://dlmf.nist.gov/8.4.E13 Encodings: TeX, pMML, png See also: Annotations for 8.4 and 8
 8.4.14 $Q\left(n+\tfrac{1}{2},z^{2}\right)=\operatorname{erfc}\left(z\right)+\frac{e^{% -z^{2}}}{\sqrt{\pi}}\sum_{k=1}^{n}\frac{z^{2k-1}}{{\left(\tfrac{1}{2}\right)_{% k}}},$
 8.4.15 $\Gamma\left(-n,z\right)=\frac{(-1)^{n}}{n!}\left(E_{1}\left(z\right)-e^{-z}% \sum_{k=0}^{n-1}\frac{(-1)^{k}k!}{z^{k+1}}\right)=\frac{(-1)^{n}}{n!}\left(% \psi\left(n+1\right)-\ln z\right)-z^{-n}\sum_{\begin{subarray}{c}k=0\\ k\neq n\end{subarray}}^{\infty}\frac{(-z)^{k}}{k!(k-n)}.$