5 Gamma Function Properties 5.8 Infinite Products 5.10 Continued Fractions

§5.9 Integral Representations

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Keywords:: gamma function, integral representations
Permalink:: http://dlmf.nist.gov/5.9
See also:: Annotations for Ch.5

§5.9(i) Gamma Function
§5.9(ii) Psi Function, Euler’s Constant, and Derivatives

§5.9(i) Gamma Function

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Notes:: For (5.9.1)–(5.9.2) see Olver (1997b, pp. 37–38). (5.9.3) follows from (5.2.1) by a change of variables. For (5.9.4)–(5.9.7), see Temme (1996b, pp. 43–45 and 77). (5.9.8) and (5.9.9) follow from (5.9.6) and (5.9.7). For (5.9.10) and (5.9.11) see Whittaker and Watson (1927, pp. 251 and 277).
Permalink:: http://dlmf.nist.gov/5.9.i
See also:: Annotations for §5.9 and Ch.5

5.9.1		$\frac{1}{\mu}\Gamma\left(\frac{\nu}{\mu}\right)\frac{1}{z^{\nu/\mu}}=\int_{0}^% {\infty}\exp\left(-zt^{\mu}\right)t^{\nu-1}\mathrm{d}t,$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral and $z$ : complex variable Referenced by: §36.2(ii), §4.10(ii), §5.9(i) Permalink: http://dlmf.nist.gov/5.9.E1 Encodings: TeX, pMML, png See also: Annotations for §5.9(i), §5.9 and Ch.5

$\Re\nu>0$ , $\mu>0$ , and $\Re z>0$ . (The fractional powers have their principal values.)

Hankel’s Loop Integral

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Keywords:: Hankel’s loop integral, gamma function
See also:: Annotations for §5.9(i), §5.9 and Ch.5

5.9.2		$\frac{1}{\Gamma\left(z\right)}=\frac{1}{2\pi i}\int_{-\infty}^{(0+)}e^{t}t^{-z% }\mathrm{d}t,$
ⓘ 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, $\int$ : integral and $z$ : complex variable A&S Ref: 6.1.4 (in a slightly different form.) Referenced by: §5.21, §5.9(i) Permalink: http://dlmf.nist.gov/5.9.E2 Encodings: TeX, pMML, png See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

where the contour begins at $-\infty$ , circles the origin once in the positive direction, and returns to $-\infty$ . $t^{-z}$ has its principal value where $t$ crosses the positive real axis, and is continuous. See Figure 5.9.1.

See accompanying text — Figure 5.9.1: $t$ -plane. Contour for Hankel’s loop integral. Magnify

5.9.3		$c^{-z}\Gamma\left(z\right)=\int_{-\infty}^{\infty}\|t\|^{2z-1}e^{-ct^{2}}\mathrm% {d}t,$
$c>0$ , $\Re z>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and $z$ : complex variable Referenced by: §5.9(i) Permalink: http://dlmf.nist.gov/5.9.E3 Encodings: TeX, pMML, png See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

where the path is the real axis.

5.9.4		$\Gamma\left(z\right)=\int_{1}^{\infty}t^{z-1}e^{-t}\mathrm{d}t+\sum_{k=0}^{% \infty}\frac{(-1)^{k}}{(z+k)k!},$
$z\neq 0,-1,-2,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $k$ : nonnegative integer and $z$ : complex variable Referenced by: §5.9(i) Permalink: http://dlmf.nist.gov/5.9.E4 Encodings: TeX, pMML, png See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

5.9.5		$\Gamma\left(z\right)=\int_{0}^{\infty}t^{z-1}\left(e^{-t}-\sum_{k=0}^{n}\frac{% (-1)^{k}t^{k}}{k!}\right)\mathrm{d}t,$
$-n-1<\Re z<-n$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\Re$ : real part, $n$ : nonnegative integer, $k$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/5.9.E5 Encodings: TeX, pMML, png See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

5.9.6	$\displaystyle\Gamma\left(z\right)\cos\left(\tfrac{1}{2}\pi z\right)$	$\displaystyle=\int_{0}^{\infty}t^{z-1}\cos t\mathrm{d}t,$
$0<\Re z<1$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part and $z$ : complex variable Referenced by: §5.9(i) Permalink: http://dlmf.nist.gov/5.9.E6 Encodings: TeX, pMML, png See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5
5.9.7	$\displaystyle\Gamma\left(z\right)\sin\left(\tfrac{1}{2}\pi z\right)$	$\displaystyle=\int_{0}^{\infty}t^{z-1}\sin t\mathrm{d}t,$
$-1<\Re z<1$ .
ⓘ 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 $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function and $z$ : complex variable Referenced by: §5.9(i) Permalink: http://dlmf.nist.gov/5.9.E7 Encodings: TeX, pMML, png See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

5.9.8		$\Gamma\left(1+\frac{1}{n}\right)\cos\left(\frac{\pi}{2n}\right)=\int_{0}^{% \infty}\cos\left(t^{n}\right)\mathrm{d}t,$
$n=2,3,4,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $n$ : nonnegative integer Referenced by: §5.9(i) Permalink: http://dlmf.nist.gov/5.9.E8 Encodings: TeX, pMML, png See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

5.9.9		$\Gamma\left(1+\frac{1}{n}\right)\sin\left(\frac{\pi}{2n}\right)=\int_{0}^{% \infty}\sin\left(t^{n}\right)\mathrm{d}t,$
$n=2,3,4,\dots$ .
ⓘ 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 $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $n$ : nonnegative integer Referenced by: §5.9(i) Permalink: http://dlmf.nist.gov/5.9.E9 Encodings: TeX, pMML, png See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

Binet’s Formula

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Keywords:: Binet’s formula, gamma function, integral representations, logarithm
See also:: Annotations for §5.9(i), §5.9 and Ch.5

5.9.10		$\operatorname{Ln}\Gamma\left(z\right)=\left(z-\tfrac{1}{2}\right)\ln z-z+% \tfrac{1}{2}\ln\left(2\pi\right)+2\int_{0}^{\infty}\frac{\operatorname{arctan}% \left(t/z\right)}{e^{2\pi t}-1}\mathrm{d}t,$
ⓘ 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, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 6.1.50 Referenced by: §5.9(i), §5.9(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8) Permalink: http://dlmf.nist.gov/5.9.E10 Encodings: TeX, pMML, png Addition (effective with 1.0.10): To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\operatorname{Ln}\Gamma\left(z\right)$ , where $\operatorname{Ln}$ is the general logarithm. Originally $\ln\Gamma\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm. Suggested 2015-02-13 by Philippe Spindel See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

where $|\operatorname{ph}z|<\pi/2$ and the inverse tangent has its principal value.

5.9.11		$\operatorname{Ln}\Gamma\left(z+1\right)=-\gamma z-\frac{1}{2\pi i}\int_{-c-% \infty i}^{-c+\infty i}\frac{\pi z^{-s}}{s\sin\left(\pi s\right)}\zeta\left(-s% \right)\mathrm{d}s,$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, $s$ : real or complex variable and $z$ : complex variable Referenced by: §5.9(i), §5.9(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8) Permalink: http://dlmf.nist.gov/5.9.E11 Encodings: TeX, pMML, png Addition (effective with 1.0.10): To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\operatorname{Ln}\Gamma\left(z+1\right)$ , where $\operatorname{Ln}$ is the general logarithm. Originally $\ln\Gamma\left(z+1\right)$ was used, where $\ln$ is the principal branch of the logarithm. Suggested 2015-02-13 by Philippe Spindel See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

where $|\operatorname{ph}z|\leq\pi-\delta$ ( $<\pi$ ), $1<c<2$ , and $\zeta\left(s\right)$ is as in Chapter 25.

For additional representations see Whittaker and Watson (1927, §§12.31–12.32).

§5.9(ii) Psi Function, Euler’s Constant, and Derivatives

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Keywords:: Euler’s constant, for derivatives, gamma function, integral representations, psi function
Notes:: See Olver (1997b, pp. 39–40), Temme (1996b, pp. 55–56), and Whittaker and Watson (1927, pp. 246–251). (5.9.15) and (5.9.17) are the differentiated forms of (5.9.10) and (5.9.11). (5.9.19) is the differentiated form of (5.2.1).
Permalink:: http://dlmf.nist.gov/5.9.ii
See also:: Annotations for §5.9 and Ch.5

For $\Re z>0$ ,

5.9.12		$\psi\left(z\right)=\int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^% {-t}}\right)\mathrm{d}t,$
ⓘ Symbols: $\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $z$ : complex variable A&S Ref: 6.3.21 Permalink: http://dlmf.nist.gov/5.9.E12 Encodings: TeX, pMML, png See also: Annotations for §5.9(ii), §5.9 and Ch.5

5.9.13		$\psi\left(z\right)=\ln z+\int_{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}% \right)e^{-tz}\mathrm{d}t,$
ⓘ Symbols: $\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable Permalink: http://dlmf.nist.gov/5.9.E13 Encodings: TeX, pMML, png See also: Annotations for §5.9(ii), §5.9 and Ch.5

5.9.14		$\psi\left(z\right)=\int_{0}^{\infty}\left(e^{-t}-\frac{1}{(1+t)^{z}}\right)% \frac{\mathrm{d}t}{t},$
ⓘ Symbols: $\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $z$ : complex variable A&S Ref: 6.3.21 Permalink: http://dlmf.nist.gov/5.9.E14 Encodings: TeX, pMML, png See also: Annotations for §5.9(ii), §5.9 and Ch.5

5.9.15		$\psi\left(z\right)=\ln z-\frac{1}{2z}-2\int_{0}^{\infty}\frac{t\mathrm{d}t}{(t% ^{2}+z^{2})(e^{2\pi t}-1)}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 6.3.21 Referenced by: §5.9(ii) Permalink: http://dlmf.nist.gov/5.9.E15 Encodings: TeX, pMML, png See also: Annotations for §5.9(ii), §5.9 and Ch.5

5.9.16		$\psi\left(z\right)+\gamma=\int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}% \mathrm{d}t=\int_{0}^{1}\frac{1-t^{z-1}}{1-t}\mathrm{d}t.$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $z$ : complex variable A&S Ref: 6.3.22 Permalink: http://dlmf.nist.gov/5.9.E16 Encodings: TeX, pMML, png See also: Annotations for §5.9(ii), §5.9 and Ch.5

5.9.17		$\psi\left(z+1\right)=-\gamma+\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}% \frac{\pi z^{-s-1}}{\sin\left(\pi s\right)}\zeta\left(-s\right)\mathrm{d}s,$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $s$ : real or complex variable and $z$ : complex variable Referenced by: §5.9(ii) Permalink: http://dlmf.nist.gov/5.9.E17 Encodings: TeX, pMML, png See also: Annotations for §5.9(ii), §5.9 and Ch.5

where $|\operatorname{ph}z|\leq\pi-\delta(<\pi)$ and $1<c<2$ .

5.9.18		$\gamma=-\int_{0}^{\infty}e^{-t}\ln t\mathrm{d}t=\int_{0}^{\infty}\left(\frac{1% }{1+t}-e^{-t}\right)\frac{\mathrm{d}t}{t}=\int_{0}^{1}(1-e^{-t})\frac{\mathrm{% d}t}{t}-\int_{1}^{\infty}e^{-t}\frac{\mathrm{d}t}{t}=\int_{0}^{\infty}\left(% \frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)\mathrm{d}t.$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\ln\NVar{z}$ : principal branch of logarithm function A&S Ref: 6.3.22 Referenced by: (9.12.31) Permalink: http://dlmf.nist.gov/5.9.E18 Encodings: TeX, pMML, png See also: Annotations for §5.9(ii), §5.9 and Ch.5

5.9.19		${\Gamma}^{(n)}\left(z\right)=\int_{0}^{\infty}(\ln t)^{n}e^{-t}t^{z-1}\mathrm{% d}t,$
$n\geq 0$ , $\Re z>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\Re$ : real part, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §5.9(ii) Permalink: http://dlmf.nist.gov/5.9.E19 Encodings: TeX, pMML, png See also: Annotations for §5.9(ii), §5.9 and Ch.5

§5.9 Integral Representations

Contents

§5.9(i) Gamma Function

Hankel’s Loop Integral

Binet’s Formula

§5.9(ii) Psi Function, Euler’s Constant, and Derivatives