# §5.9 Integral Representations

## §5.9(i) Gamma Function

 5.9.1 $\frac{1}{\mu}\Gamma\left(\frac{\nu}{\mu}\right)\frac{1}{z^{\nu/\mu}}=\int_{0}^% {\infty}\exp\left(-zt^{\mu}\right)t^{\nu-1}\,\mathrm{d}t,$

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

### Hankel’s Loop Integral

 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.

 5.9.2_5 $\frac{1}{\Gamma\left(z\right)}=\frac{{\mathrm{e}}^{z}z^{1-z}}{2\pi}\int_{-\pi}% ^{\pi}{\mathrm{e}}^{-z\Phi(t)}\,\mathrm{d}t,$ $\Re z>0$, ⓘ 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, $\Re$: real part, $z$: complex variable and $\Phi(t)$: function Source: Temme (1996b, (3.38), p. 71) Referenced by: §5.9(i), Erratum (V1.1.7) for Additions Permalink: http://dlmf.nist.gov/5.9.E2_5 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

where $\Phi(t)=1-t\cot t+\ln\left(\frac{t}{\sin t}\right)$.

 5.9.3 $c^{-z}\Gamma\left(z\right)=\int_{-\infty}^{\infty}|t|^{2z-1}e^{-ct^{2}}\,% \mathrm{d}t,$ $c>0$, $\Re z>0$,

where the path is the real axis.

 5.9.4 $\Gamma\left(z\right)=\int_{1}^{\infty}t^{z-1}e^{-t}\,\mathrm{d}t+\sum_{k=0}^{% \infty}\frac{(-1)^{k}}{(z+k)k!},$ $z\neq 0,-1,-2,\dots$.
 5.9.5 $\Gamma\left(z\right)=\int_{0}^{\infty}t^{z-1}\left(e^{-t}-\sum_{k=0}^{n}\frac{% (-1)^{k}t^{k}}{k!}\right)\,\mathrm{d}t,$ $-n-1<\Re z<-n$.
 5.9.6 $\displaystyle\Gamma\left(z\right)\cos\left(\tfrac{1}{2}\pi z\right)$ $\displaystyle=\int_{0}^{\infty}t^{z-1}\cos t\,\mathrm{d}t,$ $0<\Re z<1$, 5.9.7 $\displaystyle\Gamma\left(z\right)\sin\left(\tfrac{1}{2}\pi z\right)$ $\displaystyle=\int_{0}^{\infty}t^{z-1}\sin t\,\mathrm{d}t,$ $-1<\Re z<1$.
 5.9.8 $\Gamma\left(1+\frac{1}{n}\right)\cos\left(\frac{\pi}{2n}\right)=\int_{0}^{% \infty}\cos\left(t^{n}\right)\,\mathrm{d}t,$ $n=2,3,4,\dots$,
 5.9.9 $\Gamma\left(1+\frac{1}{n}\right)\sin\left(\frac{\pi}{2n}\right)=\int_{0}^{% \infty}\sin\left(t^{n}\right)\,\mathrm{d}t,$ $n=2,3,4,\dots$.

### Binet’s Formula

 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: (5.9.10_1), §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. Two alternative versions of Binet’s formula are

 5.9.10_1 $\operatorname{Ln}\Gamma\left(z\right)=\left(z-\tfrac{1}{2}\right)\ln z-z+% \tfrac{1}{2}\ln\left(2\pi\right)-\frac{z}{\pi}\int_{0}^{\infty}\frac{\ln\left(% 1-{\mathrm{e}}^{-2\pi t}\right)}{t^{2}+z^{2}}\,\mathrm{d}t,$ ⓘ 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, $\operatorname{Ln}\NVar{z}$: general logarithm function, $\ln\NVar{z}$: principal branch of logarithm function and $z$: complex variable Proof sketch: Can be obtained from (5.9.10) via integration by parts. Referenced by: §5.9(i), Erratum (V1.1.7) for Additions Permalink: http://dlmf.nist.gov/5.9.E10_1 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5
 5.9.10_2 $\operatorname{Ln}\Gamma\left(z\right)=\left(z-\tfrac{1}{2}\right)\ln z-z+% \tfrac{1}{2}\ln\left(2\pi\right)+\int_{0}^{\infty}{\mathrm{e}}^{-zt}\left(% \frac{1}{{\mathrm{e}}^{t}-1}-\frac{1}{t}+\frac{1}{2}\right)\frac{\,\mathrm{d}t% }{t},$ ⓘ 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, $\operatorname{Ln}\NVar{z}$: general logarithm function, $\ln\NVar{z}$: principal branch of logarithm function and $z$: complex variable Source: Whittaker and Watson (1927, §12.31) Referenced by: §5.9(i), Erratum (V1.1.7) for Additions Permalink: http://dlmf.nist.gov/5.9.E10_2 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

where $|\operatorname{ph}z|<\pi/2$.

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

 5.9.11_1 $\Gamma^{*}\left(z\right)=1-\frac{1}{2\pi\mathrm{i}}\int_{0}^{\infty}\frac{{% \mathrm{e}}^{-2\pi t}\Gamma^{*}\left(t{\mathrm{e}}^{\mathrm{i}\pi/2}\right)}{t% +\mathrm{i}z}\,\mathrm{d}t+\frac{1}{2\pi\mathrm{i}}\int_{0}^{\infty}\frac{{% \mathrm{e}}^{-2\pi t}\Gamma^{*}\left(t{\mathrm{e}}^{-\mathrm{i}\pi/2}\right)}{% t-\mathrm{i}z}\,\mathrm{d}t,$ ⓘ 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, $\int$: integral, $\Gamma^{*}\left(\NVar{z}\right)$: scaled gamma function and $z$: complex variable Source: Boyd (1994, (2.16), p. 617) Referenced by: §5.9(i), Erratum (V1.1.7) for Additions Permalink: http://dlmf.nist.gov/5.9.E11_1 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5
 5.9.11_2 $\frac{1}{\Gamma^{*}\left(z\right)}=1-\frac{1}{2\pi\mathrm{i}}\int_{0}^{\infty}% \frac{{\mathrm{e}}^{-2\pi t}\Gamma^{*}\left(t{\mathrm{e}}^{\mathrm{i}\pi/2}% \right)}{t-\mathrm{i}z}\,\mathrm{d}t+\frac{1}{2\pi\mathrm{i}}\int_{0}^{\infty}% \frac{{\mathrm{e}}^{-2\pi t}\Gamma^{*}\left(t{\mathrm{e}}^{-\mathrm{i}\pi/2}% \right)}{t+\mathrm{i}z}\,\mathrm{d}t,$ ⓘ 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, $\int$: integral, $\Gamma^{*}\left(\NVar{z}\right)$: scaled gamma function and $z$: complex variable Source: Boyd (1994, (4.2), p. 628) Referenced by: §5.9(i), Erratum (V1.1.7) for Additions Permalink: http://dlmf.nist.gov/5.9.E11_2 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

where $|\operatorname{ph}z|<\pi/2$, and the scaled gamma function $\Gamma^{*}\left(z\right)$ is defined in (5.11.3). For additional representations see Whittaker and Watson (1927, §§12.31–12.32).

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

For $\Re z>0$,

 5.9.12 $\psi\left(z\right)=\int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^% {-t}}\right)\,\mathrm{d}t,$
 5.9.13 $\psi\left(z\right)=\ln z+\int_{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}% \right)e^{-tz}\,\mathrm{d}t,$
 5.9.14 $\psi\left(z\right)=\int_{0}^{\infty}\left(e^{-t}-\frac{1}{(1+t)^{z}}\right)% \frac{\,\mathrm{d}t}{t},$
 5.9.15 $\psi\left(z\right)=\ln z-\frac{1}{2z}-2\int_{0}^{\infty}\frac{t\,\mathrm{d}t}{% (t^{2}+z^{2})(e^{2\pi t}-1)}.$
 5.9.16 $\psi\left(z\right)+\gamma=\int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\,% \mathrm{d}t=\int_{0}^{1}\frac{1-t^{z-1}}{1-t}\,\mathrm{d}t.$
 5.9.17 $\psi\left(z+1\right)=-\gamma+\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}% \frac{\pi z^{-s-1}}{\sin\left(\pi s\right)}\zeta\left(-s\right)\,\mathrm{d}s,$

where $|\operatorname{ph}z|\leq\pi-\delta$ and $1.

 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 ${\Gamma}^{(n)}\left(z\right)=\int_{0}^{\infty}(\ln t)^{n}e^{-t}t^{z-1}\,% \mathrm{d}t,$ $n\geq 0$, $\Re z>0$.