# §5.9(i) Gamma Function

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

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

# ¶ Hankel’s Loop Integral

 5.9.2 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(z\right)}=\frac{1}{2\pi i}\int_{-% \infty}^{(0+)}e^{t}t^{-z}dt,$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$: gamma function, $dx$: differential of $x$, $e$: base of exponential function, $\int$: 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

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.

 5.9.3 $c^{-z}\mathop{\Gamma\/}\nolimits\!\left(z\right)=\int_{-\infty}^{\infty}|t|^{2% z-1}e^{-ct^{2}}dt,$ $c>0$, $\realpart{z}>0$,

where the path is the real axis.

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

# ¶ Binet’s Formula

 5.9.10 $\mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(z\right)=\left(z-% \tfrac{1}{2}\right)\mathop{\ln\/}\nolimits z-z+\tfrac{1}{2}\mathop{\ln\/}% \nolimits\!\left(2\pi\right)+2\int_{0}^{\infty}\frac{\mathop{\mathrm{arctan}\/% }\nolimits\!\left(t/z\right)}{e^{2\pi t}-1}dt,$

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

 5.9.11 $\mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(z+1\right)=-% \EulerConstant z-\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{% -s}}{s\mathop{\sin\/}\nolimits\!\left(\pi s\right)}\mathop{\zeta\/}\nolimits\!% \left(-s\right)ds,$

where $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$ ($<\pi$), $1, and $\mathop{\zeta\/}\nolimits\!\left(s\right)$ 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

For $\realpart{z}>0$,

 5.9.12 $\mathop{\psi\/}\nolimits\!\left(z\right)=\int_{0}^{\infty}\left(\frac{e^{-t}}{% t}-\frac{e^{-zt}}{1-e^{-t}}\right)dt,$
 5.9.13 $\mathop{\psi\/}\nolimits\!\left(z\right)=\mathop{\ln\/}\nolimits z+\int_{0}^{% \infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}\right)e^{-tz}dt,$
 5.9.14 $\mathop{\psi\/}\nolimits\!\left(z\right)=\int_{0}^{\infty}\left(e^{-t}-\frac{1% }{(1+t)^{z}}\right)\frac{dt}{t},$
 5.9.15 $\mathop{\psi\/}\nolimits\!\left(z\right)=\mathop{\ln\/}\nolimits z-\frac{1}{2z% }-2\int_{0}^{\infty}\frac{tdt}{(t^{2}+z^{2})(e^{2\pi t}-1)}.$
 5.9.16 $\mathop{\psi\/}\nolimits\!\left(z\right)+\EulerConstant=\int_{0}^{\infty}\frac% {e^{-t}-e^{-zt}}{1-e^{-t}}dt=\int_{0}^{1}\frac{1-t^{z-1}}{1-t}dt.$
 5.9.17 $\mathop{\psi\/}\nolimits\!\left(z+1\right)=-\EulerConstant+\frac{1}{2\pi i}% \int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s-1}}{\mathop{\sin\/}\nolimits\!% \left(\pi s\right)}\mathop{\zeta\/}\nolimits\!\left(-s\right)ds,$

where $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta(<\pi)$ and $1.

 5.9.18 $\EulerConstant=-\int_{0}^{\infty}e^{-t}\mathop{\ln\/}\nolimits tdt=\int_{0}^{% \infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{dt}{t}=\int_{0}^{1}(1-e^{-t})% \frac{dt}{t}-\int_{1}^{\infty}e^{-t}\frac{dt}{t}=\int_{0}^{\infty}\left(\frac{% e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)dt.$
 5.9.19 ${\mathop{\Gamma\/}\nolimits^{(n)}}\!\left(z\right)=\int_{0}^{\infty}(\mathop{% \ln\/}\nolimits t)^{n}e^{-t}t^{z-1}dt,$ $n\geq 0$, $\realpart{z}>0$.