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

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