8 Incomplete Gamma and Related Functions Incomplete Gamma Functions 8.13 Zeros 8.15 Sums

§8.14 Integrals

ⓘ

Keywords:: incomplete gamma functions, integrals
Notes:: (8.14.1)–(8.14.2) are obtained by term-by-term integration using (8.7.1) and (8.7.3). (8.14.3)–(8.14.6) are obtained by specializing (13.10.10), (13.10.11), (13.10.3), (13.10.4) by means of (8.5.1)–(8.5.3).
Permalink:: http://dlmf.nist.gov/8.14
See also:: Annotations for Ch.8

8.14.1		$\int_{0}^{\infty}e^{-ax}\frac{\gamma\left(b,x\right)}{\Gamma\left(b\right)}\,% \mathrm{d}x=\frac{(1+a)^{-b}}{a},$
$\Re a>0$ , $\Re b>-1$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma 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, $\Re$ : real part, $x$ : real variable and $a$ : parameter Referenced by: §8.14, §8.14 Permalink: http://dlmf.nist.gov/8.14.E1 Encodings: TeX, pMML, png See also: Annotations for §8.14 and Ch.8

8.14.2		$\int_{0}^{\infty}e^{-ax}\Gamma\left(b,x\right)\,\mathrm{d}x=\Gamma\left(b% \right)\frac{1-(1+a)^{-b}}{a},$
$\Re a>-1$ , $\Re b>-1$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma 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, $\Re$ : real part, $x$ : real variable and $a$ : parameter A&S Ref: 6.5.36 Referenced by: §8.14, §8.14 Permalink: http://dlmf.nist.gov/8.14.E2 Encodings: TeX, pMML, png See also: Annotations for §8.14 and Ch.8

In (8.14.1) and (8.14.2) limiting values are used when $b=0$ .

8.14.3		$\int_{0}^{\infty}x^{a-1}\gamma\left(b,x\right)\,\mathrm{d}x=-\frac{\Gamma\left% (a+b\right)}{a},$
$\Re a<0$ , $\Re\left(a+b\right)>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part, $x$ : real variable and $a$ : parameter Referenced by: §8.14 Permalink: http://dlmf.nist.gov/8.14.E3 Encodings: TeX, pMML, png See also: Annotations for §8.14 and Ch.8

8.14.4		$\int_{0}^{\infty}x^{a-1}\Gamma\left(b,x\right)\,\mathrm{d}x=\frac{\Gamma\left(% a+b\right)}{a},$
$\Re a>0$ , $\Re\left(a+b\right)>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part, $x$ : real variable and $a$ : parameter A&S Ref: 6.5.37 Permalink: http://dlmf.nist.gov/8.14.E4 Encodings: TeX, pMML, png See also: Annotations for §8.14 and Ch.8

8.14.5		$\int_{0}^{\infty}x^{a-1}e^{-sx}\gamma\left(b,x\right)\,\mathrm{d}x=\frac{% \Gamma\left(a+b\right)}{b(1+s)^{a+b}}\*F\left(1,a+b;1+b;1/(1+s)\right),$
$\Re s>0$ , $\Re\left(a+b\right)>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric 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, $\Re$ : real part, $x$ : real variable and $a$ : parameter Permalink: http://dlmf.nist.gov/8.14.E5 Encodings: TeX, pMML, png See also: Annotations for §8.14 and Ch.8

8.14.6		$\int_{0}^{\infty}x^{a-1}e^{-sx}\Gamma\left(b,x\right)\,\mathrm{d}x=\frac{% \Gamma\left(a+b\right)}{a(1+s)^{a+b}}\*F\left(1,a+b;1+a;s/(1+s)\right),$
$\Re s>-1$ , $\Re\left(a+b\right)>0$ , $\Re a>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric 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, $\Re$ : real part, $x$ : real variable and $a$ : parameter Referenced by: §8.14 Permalink: http://dlmf.nist.gov/8.14.E6 Encodings: TeX, pMML, png See also: Annotations for §8.14 and Ch.8

For the hypergeometric function $F\left(a,b;c;z\right)$ see §15.2(i).

For additional integrals see Apelblat (1983, §8.2), Erdélyi et al. (1953b, §9.3), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2015, §6.45), Marichev (1983, pp.189–190), Oberhettinger (1972, pp. 68–69), Prudnikov et al. (1986b, §§1.2, 2.10), and Prudnikov et al. (1992a, §3.10).

8.13 Zeros 8.15 Sums

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