§8.4 Special Values

For $\mathrm{erf}\left(z\right)$, $\mathrm{erfc}\left(z\right)$, 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		$$\gamma (\frac{1}{2},z^2)=2{\int }_0^z{\mathrm{e}}^{-t^2}dt=\sqrt{\pi }\mathrm{erf}\left(z\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\int$: integral and $z$: complex variable A&S Ref: 6.5.16 Permalink: http://dlmf.nist.gov/8.4.E1 Encodings: TeX, pMML, png See also: Annotations for §8.4 and Ch.8

8.4.2		${\gamma }^{*}(a,0)$	$={\displaystyle \frac{1}{\mathrm{\Gamma }\left(a+1\right)}},$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\gamma }^{*}(a,z)$: 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 Ch.8
8.4.3		${\gamma }^{*}(\frac{1}{2},-z^2)$	$={\displaystyle \frac{2{\mathrm{e}}^{z^2}}{z\sqrt{\pi }}}F\left(z\right).$
ⓘ Symbols: $F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{*}(a,z)$: incomplete gamma function and $z$: complex variable A&S Ref: 6.5.18 Permalink: http://dlmf.nist.gov/8.4.E3 Encodings: TeX, pMML, png See also: Annotations for §8.4 and Ch.8

8.4.4		$$\mathrm{\Gamma }(0,z)={\int }_z^{\mathrm{\infty }}{t}^{-1}{\mathrm{e}}^{-t}dt=E_1\left(z\right),$$
ⓘ Symbols: $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral and $z$: complex variable A&S Ref: 6.5.15 Permalink: http://dlmf.nist.gov/8.4.E4 Encodings: TeX, pMML, png See also: Annotations for §8.4 and Ch.8

8.4.5		$$\mathrm{\Gamma }(1,z)={\mathrm{e}}^{-z},$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: 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 Ch.8

8.4.6		$$\mathrm{\Gamma }(\frac{1}{2},z^2)=2{\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}dt=\sqrt{\pi }\mathrm{erfc}\left(z\right).$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral and $z$: complex variable A&S Ref: 6.5.17 Referenced by: §8.4 Permalink: http://dlmf.nist.gov/8.4.E6 Encodings: TeX, pMML, png See also: Annotations for §8.4 and Ch.8

For $n=0,1,2,\mathrm{\dots }$,

8.4.7		$\gamma (n+1,z)$	$=n!(1-{\mathrm{e}}^{-z}{e}_n(z)),$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\gamma (a,z)$: incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions Referenced by: §8.11(v) Permalink: http://dlmf.nist.gov/8.4.E7 Encodings: TeX, pMML, png See also: Annotations for §8.4 and Ch.8
8.4.8		$\mathrm{\Gamma }(n+1,z)$	$=n!{\mathrm{e}}^{-z}{e}_n(z),$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions Permalink: http://dlmf.nist.gov/8.4.E8 Encodings: TeX, pMML, png See also: Annotations for §8.4 and Ch.8
8.4.9		$P(n+1,z)$	$=1-{\mathrm{e}}^{-z}{e}_n(z),$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $P(a,z)$: 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 Ch.8
8.4.10		$Q(n+1,z)$	$={\mathrm{e}}^{-z}{e}_n(z),$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions Permalink: http://dlmf.nist.gov/8.4.E10 Encodings: TeX, pMML, png See also: Annotations for §8.4 and Ch.8

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 Ch.8

Also

8.4.12		$${\gamma }^{*}(-n,z)=z^n,$$
ⓘ Symbols: ${\gamma }^{*}(a,z)$: 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 Ch.8

8.4.13		$$\mathrm{\Gamma }(1-n,z)=z^{1-n}{E}_n\left(z\right),$$
ⓘ Symbols: $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: 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 Ch.8

8.4.14		$$Q(n+\frac{1}{2},z^2)=\mathrm{erfc}\left(z\right)+\frac{{\mathrm{e}}^{-z^2}}{\sqrt{\pi }}\sum _{k=1}^n\frac{z^{2k-1}}{{\left(\frac{1}{2}\right)}_k},$$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer Referenced by: §8.4 Permalink: http://dlmf.nist.gov/8.4.E14 Encodings: TeX, pMML, png See also: Annotations for §8.4 and Ch.8

8.4.15		$$\mathrm{\Gamma }(-n,z)=\frac{{(-1)}^n}{n!}\left(E_1\left(z\right)-{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{{(-1)}^{k}k!}{z^{k+1}}\right)=\frac{{(-1)}^n}{n!}\left(\psi \left(n+1\right)-\ln z\right)-z^{-n}\sum _{\begin{array}{c}\hfill k=0\hfill \\ \hfill k\ne n\hfill \end{array}}^{\mathrm{\infty }}\frac{{(-z)}^k}{k!(k-n)}.$$
ⓘ Symbols: $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\ln z$: principal branch of logarithm function, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer A&S Ref: 6.5.19 Referenced by: §8.19(iii), §8.19(iv), §8.4 Permalink: http://dlmf.nist.gov/8.4.E15 Encodings: TeX, pMML, png See also: Annotations for §8.4 and Ch.8