§8.6 Integral Representations

Permalink:: http://dlmf.nist.gov/8.6

§8.6(i) Integrals Along the Real Line
§8.6(ii) Contour Integrals
§8.6(iii) Compendia

§8.6(i) Integrals Along the Real Line

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 $\mathop{J_{{a}}\/}\nolimits\!\left(2\sqrt{zt}\right)$ 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).
Keywords:: incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.6.i

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

8.6.1 $\mathop{\gamma\/}\nolimits\!\left(a,z\right)=\frac{z^{a}}{\mathop{\sin\/}\nolimits\!\left(\pi a\right)}\int _{0}^{\pi}e^{{z\mathop{\cos\/}\nolimits t}}\mathop{\cos\/}\nolimits\!\left(at+z\mathop{\sin\/}\nolimits t\right)dt,$ $a\notin\Integer$ ,

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\Integer$ : set of all integers, $\int$ : integral, $\notin$ : not an element of, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.6(i)
Permalink:: http://dlmf.nist.gov/8.6.E1
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8.6.2 $\mathop{\gamma\/}\nolimits\!\left(a,z\right)=z^{{\frac{1}{2}a}}\int _{0}^{\infty}e^{{-t}}t^{{\frac{1}{2}a-1}}\mathop{J_{{a}}\/}\nolimits\!\left(2\sqrt{zt}\right)dt,$ $\realpart{a}>0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.6(i)
Permalink:: http://dlmf.nist.gov/8.6.E2
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8.6.3 $\mathop{\gamma\/}\nolimits\!\left(a,z\right)=z^{a}\int _{0}^{\infty}\mathop{\exp\/}\nolimits\!\left(-at-ze^{{-t}}\right)dt,$ $\realpart{a}>0$ .

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.6(i)
Permalink:: http://dlmf.nist.gov/8.6.E3
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8.6.4 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=\frac{z^{a}e^{{-z}}}{\mathop{\Gamma\/}\nolimits\!\left(1-a\right)}\int _{0}^{\infty}\frac{t^{{-a}}e^{{-t}}}{z+t}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ , $\realpart{a}<1$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.25(ii), §8.6(i)
Permalink:: http://dlmf.nist.gov/8.6.E4
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8.6.5 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=z^{a}e^{{-z}}\int _{0}^{\infty}\frac{e^{{-zt}}}{(1+t)^{{1-a}}}dt,$ $\realpart{z}>0$ ,

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.6(i)
Permalink:: http://dlmf.nist.gov/8.6.E5
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8.6.6 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=\frac{2z^{{\frac{1}{2}a}}e^{{-z}}}{\mathop{\Gamma\/}\nolimits\!\left(1-a\right)}\int _{0}^{\infty}e^{{-t}}t^{{-\frac{1}{2}a}}\mathop{K_{{a}}\/}\nolimits\!\left(2\sqrt{zt}\right)dt,$ $\realpart{a}<1$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.6(i)
Permalink:: http://dlmf.nist.gov/8.6.E6
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8.6.7 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=z^{a}\int _{0}^{\infty}\mathop{\exp\/}\nolimits\!\left(at-ze^{t}\right)dt,$ $\realpart{z}>0$ .

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.6(i)
Permalink:: http://dlmf.nist.gov/8.6.E7
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§8.6(ii) Contour Integrals

Notes:: For (8.6.8) assume temporarily $\realpart{a}>0$ , 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).
Keywords:: incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.6.ii

8.6.8 $\mathop{\gamma\/}\nolimits\!\left(a,z\right)=\frac{-iz^{a}}{2\mathop{\sin\/}\nolimits\!\left(\pi a\right)}\int _{{-1}}^{{(0+)}}t^{{a-1}}e^{{zt}}dt,$ $z\neq 0$ , $a\notin\Integer$ ;

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\Integer$ : set of all integers, $\int$ : integral, $\notin$ : not an element of, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.6(i), §8.6(ii)
Permalink:: http://dlmf.nist.gov/8.6.E8
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$t^{{a-1}}$ takes its principal value where the path intersects the positive real axis, and is continuous elsewhere on the path.

8.6.9 $\mathop{\Gamma\/}\nolimits\!\left(-a,ze^{{\pm\pi i}}\right)=\frac{e^{z}e^{{\mp\pi ia}}}{\mathop{\Gamma\/}\nolimits\!\left(1+a\right)}\int _{0}^{\infty}\frac{t^{a}e^{{-zt}}}{t-1}dt,$ $\realpart{z}>0$ , $\realpart{a}>-1$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.6(ii)
Permalink:: http://dlmf.nist.gov/8.6.E9
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where the integration path passes above or below the pole at $t=1$ , according as upper or lower signs are taken.

¶ Mellin–Barnes Integrals

Keywords:: incomplete gamma functions

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

8.6.10 $\mathop{\gamma\/}\nolimits\!\left(a,z\right)=\frac{1}{2\pi i}\int _{{c-i\infty}}^{{c+i\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(s\right)}{a-s}z^{{a-s}}ds,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ , $a\neq 0,-1,-2,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable, $a$ : parameter and $c$ : real constant
Referenced by:: ¶ ‣ §8.6(ii), §8.6(ii)
Permalink:: http://dlmf.nist.gov/8.6.E10
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8.6.11 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=\frac{1}{2\pi i}\int _{{c-i\infty}}^{{c+i\infty}}\mathop{\Gamma\/}\nolimits\!\left(s+a\right)\frac{z^{{-s}}}{s}ds,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable, $a$ : parameter and $c$ : real constant
Referenced by:: ¶ ‣ §8.19(i), ¶ ‣ §8.6(ii)
Permalink:: http://dlmf.nist.gov/8.6.E11
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8.6.12 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=-\frac{z^{{a-1}}e^{{-z}}}{\mathop{\Gamma\/}\nolimits\!\left(1-a\right)}\*\frac{1}{2\pi i}\int _{{c-i\infty}}^{{c+i\infty}}\mathop{\Gamma\/}\nolimits\!\left(s+1-a\right)\frac{\pi z^{{-s}}}{\mathop{\sin\/}\nolimits\!\left(\pi s\right)}ds,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{3}{2}\pi$ , $a\neq 1,2,3,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable, $a$ : parameter and $c$ : real constant
Referenced by:: ¶ ‣ §8.19(i), ¶ ‣ §8.6(ii), §8.6(ii)
Permalink:: http://dlmf.nist.gov/8.6.E12
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§8.6(iii) Compendia

Keywords:: incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.6.iii

For collections of integral representations of $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ and $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ 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