§8.6 Integral Representations

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Permalink:: http://dlmf.nist.gov/8.6
See also:: Annotations for Ch.8

§8.6(i) Integrals Along the Real Line

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Keywords:: along real line, incomplete gamma functions, integral representations
Notes:: For (8.6.1) use (8.6.8), replacing $t$ by $e^{it}$ with $-\pi\leq t\leq\pi$ . For (8.6.2) substitute for $J_{a}\left(2\sqrt{zt}\right)$ by (10.2.2), integrate term by term and refer to (8.7.1) and (8.2.6). (8.6.6) may be proved in a similar manner with the aid also of (10.25.2), (10.27.4), (8.2.3), and analytic continuation when $a=-2,-3,-4,\dots$ . For (8.6.3) and (8.6.7) apply (8.2.1) and (8.2.2), taking new integration variables $ze^{\mp t}$ . For (8.6.4) and (8.6.5) see Temme (1996b, §§11.2.1–11.2.2).
Permalink:: http://dlmf.nist.gov/8.6.i
See also:: Annotations for §8.6 and Ch.8

For the Bessel function $J_{\nu}\left(z\right)$ and modified Bessel function $K_{\nu}\left(z\right)$ , see §§10.2(ii) and 10.25(ii).

8.6.1	$\displaystyle\gamma\left(a,z\right)$	$\displaystyle=\frac{z^{a}}{\sin\left(\pi a\right)}\int_{0}^{\pi}e^{z\cos t}% \cos\left(at+z\sin t\right)\,\mathrm{d}t,$
$a\notin\mathbb{Z}$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\mathbb{Z}$ : set of all integers, $\int$ : integral, $\notin$ : not an element of, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $a$ : parameter Referenced by: §8.6(i) Permalink: http://dlmf.nist.gov/8.6.E1 Encodings: TeX, pMML, png See also: Annotations for §8.6(i), §8.6 and Ch.8
8.6.2	$\displaystyle\gamma\left(a,z\right)$	$\displaystyle=z^{\frac{1}{2}a}\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}a-1}J_{a}% \left(2\sqrt{zt}\right)\,\mathrm{d}t,$
$\Re a>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $a$ : parameter Referenced by: §8.6(i) Permalink: http://dlmf.nist.gov/8.6.E2 Encodings: TeX, pMML, png See also: Annotations for §8.6(i), §8.6 and Ch.8
8.6.3	$\displaystyle\gamma\left(a,z\right)$	$\displaystyle=z^{a}\int_{0}^{\infty}\exp\left(-at-ze^{-t}\right)\,\mathrm{d}t,$
$\Re a>0$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $a$ : parameter Referenced by: §8.6(i) Permalink: http://dlmf.nist.gov/8.6.E3 Encodings: TeX, pMML, png See also: Annotations for §8.6(i), §8.6 and Ch.8

8.6.4	$\displaystyle\Gamma\left(a,z\right)$	$\displaystyle=\frac{z^{a}e^{-z}}{\Gamma\left(1-a\right)}\int_{0}^{\infty}\frac% {t^{-a}e^{-t}}{z+t}\,\mathrm{d}t,$
$\|\operatorname{ph}z\|<\pi$ , $\Re a<1$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $z$ : complex variable and $a$ : parameter Referenced by: §8.25(ii), §8.6(i) Permalink: http://dlmf.nist.gov/8.6.E4 Encodings: TeX, pMML, png See also: Annotations for §8.6(i), §8.6 and Ch.8
8.6.5	$\displaystyle\Gamma\left(a,z\right)$	$\displaystyle=z^{a}e^{-z}\int_{0}^{\infty}\frac{e^{-zt}}{(1+t)^{1-a}}\,\mathrm% {d}t,$
$\Re z>0$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $a$ : parameter Referenced by: §8.6(i), (8.7.6) Permalink: http://dlmf.nist.gov/8.6.E5 Encodings: TeX, pMML, png See also: Annotations for §8.6(i), §8.6 and Ch.8
8.6.6	$\displaystyle\Gamma\left(a,z\right)$	$\displaystyle=\frac{2z^{\frac{1}{2}a}e^{-z}}{\Gamma\left(1-a\right)}\int_{0}^{% \infty}e^{-t}t^{-\frac{1}{2}a}K_{a}\left(2\sqrt{zt}\right)\,\mathrm{d}t,$
$\Re a<1$ ,
ⓘ 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, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part, $z$ : complex variable and $a$ : parameter Referenced by: §8.6(i) Permalink: http://dlmf.nist.gov/8.6.E6 Encodings: TeX, pMML, png See also: Annotations for §8.6(i), §8.6 and Ch.8
8.6.7	$\displaystyle\Gamma\left(a,z\right)$	$\displaystyle=z^{a}\int_{0}^{\infty}\exp\left(at-ze^{t}\right)\,\mathrm{d}t,$
$\Re z>0$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $a$ : parameter Referenced by: §8.6(i) Permalink: http://dlmf.nist.gov/8.6.E7 Encodings: TeX, pMML, png See also: Annotations for §8.6(i), §8.6 and Ch.8

§8.6(ii) Contour Integrals

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Keywords:: contour integrals, incomplete gamma functions, integral representations, of imaginary argument
Notes:: For (8.6.8) assume temporarily $\Re a>0$ , collapse the integration path onto the interval $[-1,0]$ and use (8.2.1). For (8.6.9) see Temme (1996a). For (8.6.10)–(8.6.12) see Paris and Kaminski (2001, §3.4.3).
Referenced by:: Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))
Permalink:: http://dlmf.nist.gov/8.6.ii
See also:: Annotations for §8.6 and Ch.8

8.6.8		$\gamma\left(a,z\right)=\frac{-\mathrm{i}z^{a}}{2\sin\left(\pi a\right)}\int_{-% 1}^{(0+)}t^{a-1}e^{zt}\,\mathrm{d}t,$
$z\neq 0$ , $a\notin\mathbb{Z}$ ;
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\mathbb{Z}$ : set of all integers, $\int$ : integral, $\notin$ : not an element of, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $a$ : parameter Referenced by: §8.6(i), §8.6(ii) Permalink: http://dlmf.nist.gov/8.6.E8 Encodings: TeX, pMML, png See also: Annotations for §8.6(ii), §8.6 and Ch.8

$t^{a-1}$ takes its principal value where the path intersects the positive real axis, and is continuous elsewhere on the path.

8.6.9		$\Gamma\left(-a,ze^{\pm\pi i}\right)=\frac{e^{z}e^{\mp\pi\mathrm{i}a}}{\Gamma% \left(1+a\right)}\int_{0}^{\infty}\frac{t^{a}e^{-zt}}{t-1}\,\mathrm{d}t,$
$\Re z>0$ , $\Re a>-1$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $a$ : parameter Referenced by: §8.6(ii) Permalink: http://dlmf.nist.gov/8.6.E9 Encodings: TeX, pMML, png See also: Annotations for §8.6(ii), §8.6 and Ch.8

where the integration path passes above or below the pole at $t=1$ , according as upper or lower signs are taken.

Mellin–Barnes Integrals

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Keywords:: Mellin–Barnes type, incomplete gamma functions, integral representations
Referenced by:: Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))
Errata (effective with 1.0.17):: The descriptions for the paths of integration of the Mellin–Barnes integrals (8.6.10)–(8.6.12) have been updated. Previously the description for (8.6.11) stated incorrectly that the path of integration had to separate the poles of the gamma function from the pole at $s=0$ .
Reported 2017-07-10 by Kurt Fischer
See also:: Annotations for §8.6(ii), §8.6 and Ch.8

In (8.6.10)–(8.6.12), $c$ is a real constant and the path of integration is indented (if necessary) so that in the case of (8.6.10) it separates the poles of the gamma function from the pole at $s=a$ , in the case of (8.6.11) it is to the right of all poles, and in the case of (8.6.12) it separates the poles of the gamma function from the poles at $s=0,1,2,\ldots$ .

8.6.10		$\gamma\left(a,z\right)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{% \Gamma\left(s\right)}{a-s}z^{a-s}\,\mathrm{d}s,$
$\|\operatorname{ph}z\|<\tfrac{1}{2}\pi$ , $a\neq 0,-1,-2,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : complex variable, $a$ : parameter and $c$ : real constant Keywords: Mellin transform Referenced by: §8.6(ii), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii)) Permalink: http://dlmf.nist.gov/8.6.E10 Encodings: TeX, pMML, png See also: Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8

8.6.11	$\displaystyle\Gamma\left(a,z\right)$	$\displaystyle=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(s+a% \right)\frac{z^{-s}}{s}\,\mathrm{d}s,$
$\|\operatorname{ph}z\|<\tfrac{1}{2}\pi$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : complex variable, $a$ : parameter and $c$ : real constant Keywords: Mellin transform Referenced by: §8.19(i), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii)) Permalink: http://dlmf.nist.gov/8.6.E11 Encodings: TeX, pMML, png See also: Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8
8.6.12	$\displaystyle\Gamma\left(a,z\right)$	$\displaystyle=-\frac{z^{a-1}e^{-z}}{\Gamma\left(1-a\right)}\*\frac{1}{2\pi i}% \int_{c-i\infty}^{c+i\infty}\Gamma\left(s+1-a\right)\frac{\pi z^{-s}}{\sin% \left(\pi s\right)}\,\mathrm{d}s,$
$\|\operatorname{ph}z\|<\tfrac{3}{2}\pi$ , $a\neq 1,2,3,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $a$ : parameter and $c$ : real constant Keywords: Mellin transform Referenced by: §8.19(i), §8.6(ii), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii)) Permalink: http://dlmf.nist.gov/8.6.E12 Encodings: TeX, pMML, png See also: Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8

§8.6(iii) Compendia

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Keywords:: compendia, incomplete gamma functions, integral representations
Permalink:: http://dlmf.nist.gov/8.6.iii
See also:: Annotations for §8.6 and Ch.8

For collections of integral representations of $\gamma\left(a,z\right)$ and $\Gamma\left(a,z\right)$ see Erdélyi et al. (1953b, §9.3), Oberhettinger (1972, pp. 68–69), Oberhettinger and Badii (1973, pp. 309–312), Prudnikov et al. (1992b, §3.10), and Temme (1996b, pp. 282–283).

§8.6 Integral Representations

Contents

§8.6(i) Integrals Along the Real Line

§8.6(ii) Contour Integrals

Mellin–Barnes Integrals

§8.6(iii) Compendia

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