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DLMF: §8.4 Special Values ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions

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8 Incomplete Gamma and Related Functions Incomplete Gamma Functions 8.3 Graphics 8.5 Confluent Hypergeometric Representations

§8.4 Special Values

ⓘ

Keywords:: incomplete gamma functions, special values
Notes:: See Erdélyi et al. (1953b, Chapter 9). (8.4.14) follows from (8.4.6) and (8.8.6). (8.4.15) follows from the first series in (8.7.3) by combining $\Gamma\left(a\right)$ with the term $k=n$ of the series, then taking the limit as $a\rightarrow-n$ .
Permalink:: http://dlmf.nist.gov/8.4
See also:: Annotations for Ch.8

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		$\gamma\left(\tfrac{1}{2},z^{2}\right)=2\int_{0}^{z}e^{-t^{2}}\,\mathrm{d}t=% \sqrt{\pi}\operatorname{erf}\left(z\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : 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		$\displaystyle\gamma^{*}\left(a,0\right)$	$\displaystyle=\frac{1}{\Gamma\left(a+1\right)},$
ⓘ 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 Ch.8
8.4.3		$\displaystyle\gamma^{*}\left(\tfrac{1}{2},-z^{2}\right)$	$\displaystyle=\frac{2e^{z^{2}}}{z\sqrt{\pi}}F\left(z\right).$
ⓘ Symbols: $F\left(\NVar{z}\right)$ : Dawson’s integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : 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		$\Gamma\left(0,z\right)=\int_{z}^{\infty}t^{-1}e^{-t}\,\mathrm{d}t=E_{1}\left(z% \right),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : 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		$\Gamma\left(1,z\right)=e^{-z},$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : 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 Ch.8

8.4.6		$\Gamma\left(\tfrac{1}{2},z^{2}\right)=2\int_{z}^{\infty}e^{-t^{2}}\,\mathrm{d}% t=\sqrt{\pi}\operatorname{erfc}\left(z\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : 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 $n=0,1,2,\dots$ ,

8.4.7		$\displaystyle\gamma\left(n+1,z\right)$	$\displaystyle=n!(1-e^{-z}e_{n}(z)),$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\gamma\left(\NVar{a},\NVar{z}\right)$ : 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		$\displaystyle\Gamma\left(n+1,z\right)$	$\displaystyle=n!e^{-z}e_{n}(z),$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\Gamma\left(\NVar{a},\NVar{z}\right)$ : 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		$\displaystyle P\left(n+1,z\right)$	$\displaystyle=1-e^{-z}e_{n}(z),$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $P\left(\NVar{a},\NVar{z}\right)$ : 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 Ch.8
8.4.10		$\displaystyle Q\left(n+1,z\right)$	$\displaystyle=e^{-z}e_{n}(z),$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $Q\left(\NVar{a},\NVar{z}\right)$ : 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		$\gamma^{*}\left(-n,z\right)=z^{n},$
ⓘ 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 Ch.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 Ch.8

8.4.14		$Q\left(n+\tfrac{1}{2},z^{2}\right)=\operatorname{erfc}\left(z\right)+\frac{e^{% -z^{2}}}{\sqrt{\pi}}\sum_{k=1}^{n}\frac{z^{2k-1}}{{\left(\tfrac{1}{2}\right)_{% k}}},$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm, $Q\left(\NVar{a},\NVar{z}\right)$ : 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		$\Gamma\left(-n,z\right)=\frac{(-1)^{n}}{n!}\left(E_{1}\left(z\right)-e^{-z}% \sum_{k=0}^{n-1}\frac{(-1)^{k}k!}{z^{k+1}}\right)=\frac{(-1)^{n}}{n!}\left(% \psi\left(n+1\right)-\ln z\right)-z^{-n}\sum_{\begin{subarray}{c}k=0\\ k\neq n\end{subarray}}^{\infty}\frac{(-z)^{k}}{k!(k-n)}.$
ⓘ Symbols: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $!$ : factorial (as in $n!$ ), $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\ln\NVar{z}$ : 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

8.3 Graphics 8.5 Confluent Hypergeometric Representations

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