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

§8.14 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).
Keywords:: incomplete gamma functions, integrals
Permalink:: http://dlmf.nist.gov/8.14

8.14.1 $\int_{0}^{\infty}e^{{-ax}}\frac{\mathop{\gamma\/}\nolimits\!\left(b,x\right)}{% \mathop{\Gamma\/}\nolimits\!\left(b\right)}dx=\frac{(1+a)^{{-b}}}{a},$ $\realpart{a}>0$ , $\realpart{b}>-1$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part, : real variable and : parameter
Referenced by:: §8.14, §8.14
Permalink:: http://dlmf.nist.gov/8.14.E1
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8.14.2 $\int_{0}^{\infty}e^{{-ax}}\mathop{\Gamma\/}\nolimits\!\left(b,x\right)dx=% \mathop{\Gamma\/}\nolimits\!\left(b\right)\frac{1-(1+a)^{{-b}}}{a},$ $\realpart{a}>-1$ , $\realpart{b}>-1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part, : real variable and : parameter
A&S Ref:: 6.5.36
Referenced by:: §8.14, §8.14
Permalink:: http://dlmf.nist.gov/8.14.E2
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In (8.14.1) and (8.14.2) limiting values are used when .

8.14.3 $\int_{0}^{\infty}x^{{a-1}}\mathop{\gamma\/}\nolimits\!\left(b,x\right)dx=-% \frac{\mathop{\Gamma\/}\nolimits\!\left(a+b\right)}{a},$ $\realpart{a}<0$ , $\realpart{(a+b)}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part, : real variable and : parameter
Referenced by:: §8.14
Permalink:: http://dlmf.nist.gov/8.14.E3
Encodings:: TeX, pMML, png

8.14.4 $\int_{0}^{\infty}x^{{a-1}}\mathop{\Gamma\/}\nolimits\!\left(b,x\right)dx=\frac% {\mathop{\Gamma\/}\nolimits\!\left(a+b\right)}{a},$ $\realpart{a}>0$ , $\realpart{(a+b)}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part, : real variable and : parameter
A&S Ref:: 6.5.37
Permalink:: http://dlmf.nist.gov/8.14.E4
Encodings:: TeX, pMML, png

8.14.5 $\int_{0}^{\infty}x^{{a-1}}e^{{-sx}}\mathop{\gamma\/}\nolimits\!\left(b,x\right% )dx=\frac{\mathop{\Gamma\/}\nolimits\!\left(a+b\right)}{b(1+s)^{{a+b}}}\*% \mathop{F\/}\nolimits\!\left(1,a+b;1+b;1/(1+s)\right),$ $\realpart{s}>0$ , $\realpart{(a+b)}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : differential of , : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part, : real variable and : parameter
Permalink:: http://dlmf.nist.gov/8.14.E5
Encodings:: TeX, pMML, png

8.14.6 $\int_{0}^{\infty}x^{{a-1}}e^{{-sx}}\mathop{\Gamma\/}\nolimits\!\left(b,x\right% )dx=\frac{\mathop{\Gamma\/}\nolimits\!\left(a+b\right)}{a(1+s)^{{a+b}}}\*% \mathop{F\/}\nolimits\!\left(1,a+b;1+a;s/(1+s)\right),$ $\realpart{s}>-1$ , $\realpart{(a+b)}>0$ , $\realpart{a}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : differential of , : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part, : real variable and : parameter
Referenced by:: §8.14
Permalink:: http://dlmf.nist.gov/8.14.E6
Encodings:: TeX, pMML, png

For the hypergeometric function $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ 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 (2000, §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).