# §8.4 Special Values

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

 8.4.1 $\mathop{\gamma\/}\nolimits\!\left(\tfrac{1}{2},z^{2}\right)=2\int_{0}^{z}e^{-t% ^{2}}\mathrm{d}t=\sqrt{\pi}\mathop{\mathrm{erf}\/}\nolimits\!\left(z\right),$
 8.4.2 $\displaystyle\mathop{\gamma^{*}\/}\nolimits\!\left(a,0\right)$ $\displaystyle=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(a+1\right)},$ 8.4.3 $\displaystyle\mathop{\gamma^{*}\/}\nolimits\!\left(\tfrac{1}{2},-z^{2}\right)$ $\displaystyle=\frac{2e^{z^{2}}}{z\sqrt{\pi}}\mathop{F\/}\nolimits\!\left(z% \right).$
 8.4.4 $\mathop{\Gamma\/}\nolimits\!\left(0,z\right)=\int_{z}^{\infty}t^{-1}e^{-t}% \mathrm{d}t=\mathop{E_{1}\/}\nolimits\!\left(z\right),$
 8.4.5 $\mathop{\Gamma\/}\nolimits\!\left(1,z\right)=e^{-z},$ Symbols: $\mathrm{e}$: base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: 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
 8.4.6 $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2},z^{2}\right)=2\int_{z}^{\infty}% e^{-t^{2}}\mathrm{d}t=\sqrt{\pi}\mathop{\mathrm{erfc}\/}\nolimits\!\left(z% \right).$

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

 8.4.7 $\displaystyle\mathop{\gamma\/}\nolimits\!\left(n+1,z\right)$ $\displaystyle=n!(1-e^{-z}e_{n}(z)),$ 8.4.8 $\displaystyle\mathop{\Gamma\/}\nolimits\!\left(n+1,z\right)$ $\displaystyle=n!e^{-z}e_{n}(z),$ 8.4.9 $\displaystyle\mathop{P\/}\nolimits\!\left(n+1,z\right)$ $\displaystyle=1-e^{-z}e_{n}(z),$ 8.4.10 $\displaystyle\mathop{Q\/}\nolimits\!\left(n+1,z\right)$ $\displaystyle=e^{-z}e_{n}(z),$

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

Also

 8.4.12 $\mathop{\gamma^{*}\/}\nolimits\!\left(-n,z\right)=z^{n},$ Symbols: $\mathop{\gamma^{*}\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: 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
 8.4.13 $\mathop{\Gamma\/}\nolimits\!\left(1-n,z\right)=z^{1-n}\mathop{E_{n}\/}% \nolimits\!\left(z\right),$
 8.4.14 $\mathop{Q\/}\nolimits\!\left(n+\tfrac{1}{2},z^{2}\right)=\mathop{\mathrm{erfc}% \/}\nolimits\!\left(z\right)+\frac{e^{-z^{2}}}{\sqrt{\pi}}\sum_{k=1}^{n}\frac{% z^{2k-1}}{{\left(\tfrac{1}{2}\right)_{k}}},$
 8.4.15 $\mathop{\Gamma\/}\nolimits\!\left(-n,z\right)=\frac{(-1)^{n}}{n!}\left(\mathop% {E_{1}\/}\nolimits\!\left(z\right)-e^{-z}\sum_{k=0}^{n-1}\frac{(-1)^{k}k!}{z^{% k+1}}\right)=\frac{(-1)^{n}}{n!}\left(\mathop{\psi\/}\nolimits\!\left(n+1% \right)-\mathop{\ln\/}\nolimits z\right)-z^{-n}\sum_{\begin{subarray}{c}k=0\\ k\neq n\end{subarray}}^{\infty}\frac{(-z)^{k}}{k!(k-n)}.$