Symbols:Gamma

Gamma

Gamma Function

$\Gamma \left({z}\right)$

The Gamma function $\Gamma: \C \to \C \ $ is defined, for the open right half-plane, as:

$\displaystyle \Gamma \left({z}\right) = \int_0^{\to \infty} t^{z-1} e^{-t} \ \mathrm d t$

and for all other values of $z$ except the non-positive integers as:

$\Gamma \left({z + 1}\right) = z \Gamma \left({z}\right)$

Other equivalent definitions exist, as follows.

Weierstrass Form

Of note is the Weierstrass form:

$\displaystyle \frac 1 {\Gamma \left({z}\right)} = z e^{\gamma z} \prod_{n=1}^\infty \left({\left({ 1 + \frac z n}\right) e^{\frac {-z} n}}\right)$

where $\gamma$ is the Euler-Mascheroni constant. The Weierstrass expression is valid for all $\C$.

Euler Form

Another important form of the Gamma function is the Euler form:

$\displaystyle \Gamma \left({z}\right) = \frac 1 z \prod_{n=1}^\infty \left({ \left({1 + \frac 1 n}\right)^z \left({1 + \frac z n}\right)^{-1}}\right) = \lim_{m \to \infty} \frac {m^z m!} {z \left({z+1}\right) \left({z+2}\right) \ldots \left({z+m}\right)}$

which is valid except for $z \in \left\{{0, -1, -2, \ldots}\right\} \ $.

The $\LaTeX$ code for $\Gamma \left({z}\right)$ is \Gamma \left({z}\right).

The Euler-Mascheroni Constant

$\gamma$

The Euler-Mascheroni Constant $\gamma$ is the real number that is defined as:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \gamma\)	\(:=\)	\(\displaystyle \lim_{n \to +\infty} \left({\sum_{k=1}^n \frac 1 k - \int_1^n \frac 1 x \ \mathrm dx}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \lim_{n \to +\infty} \left({H_n - \ln n}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

where $H_n$ is the harmonic series and $\ln$ is the natural logarithm.

The existence of this constant is demonstrated in Existence of Euler-Mascheroni Constant.

Its value is approximately $0.57721\ 56649\ 01532\ 86060\ 6512 \ldots$ This sequence is A001620 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The $\LaTeX$ code for $\gamma$ is \gamma.