# Eikonal Blog

## 2010.01.04

### Gamma Function

Definitions:

• 1) Euler: $\Gamma(z) = \int_0^\infty e^{-t} t^{z-1} dt$, ($\Re z > 0$);
• 2) Gauss: $\Gamma(z) = \lim_{n\rightarrow\infty} \frac{n! n^z}{(z)_{n+1}}$, ($z\not\in{\Bbb N}_0$)
• 3) Weierstrass: $\Gamma(z)^{-1} =z e^{\gamma z} \prod_{n=1}^\infty (1+\frac{z}{n}) e^{-\frac{z}{n}}$, where $\gamma :\equiv \lim_{n\rightarrow \infty} (\sum_{k=1}^n \frac1{k} - \ln(n+1)) \sim 0.577,215,7$ is the Euler-Mascheroni constant.

The Mittag-Leffler expansion:

$\Gamma(z) = \sum_{n=0}^\infty \frac{(-)^n}{n!} \frac1{n+z} + \int_{1}^\infty e^{-t} t^{z-1} dt.$

It indicates that $\Gamma(-n+\epsilon) \sim \frac{(-)^n}{n!} \frac1\epsilon$.

Properties:

• $\Gamma(z+1)=z\Gamma(z)$.
• $\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$, ($z\not\in{\Bbb Z}$).
• $\Gamma(\frac12)=\sqrt{\pi}$.
• The multiplication theorem: $\prod_{n=0}^{m-1}\Gamma(z+\frac{n}{m}) = (2\pi)^{\frac{m-1}2} m^{\frac12 - mz} \Gamma(mz)$.
• The duplication formula: $\Gamma(x)\Gamma(z+\frac12) = 2^{1-2z} \sqrt{\pi} \Gamma(2z)$.

Hankel representations:

• $\Gamma(z) = \frac1{e^{2\pi i z}-1} \int_C e^{-\zeta}\zeta^{z-1}d\zeta$,
• $\Gamma(z) = -\frac1{2i\sin(\pi z)} \int_C e^{-\zeta}(-\zeta)^{z-1}d\zeta$,
• $\Gamma(z)^{-1} = -\frac1{2\pi} \int_C e^{-\zeta}\zeta^{-z}d\zeta$.

Here the contour $C$ comes from plus infinity narrowly above the $x$ axis, circles once around origin and returns to plus infinity narrowly below the real axis.

Stirling formula:

$\Gamma(z) = \sqrt{\frac{2\pi}{z}} (\frac{z}{e})^z [1+\frac1{12 z}+\cdots]$

Misc:

• $\lim_{n\rightarrow\infty} \int_0^{n} [e^{-t}-(1-\frac{t}{n})^n] t^{z-1} dt = 0$, $\Re z > 0$.
• $|\Gamma(iy)|^2 = \frac{\pi}{y \sinh(\pi y)}$

Source: “The Functions of Mathematical Physics” by Harry Hochstadt