From ProofWiki
Jump to: navigation, search

Previous  ... Next


The $3$rd letter of the Greek alphabet.

Minuscule: $\gamma$
Majuscule: $\Gamma$

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

The $\LaTeX$ code for \(\Gamma\) is \Gamma .

Gamma Function

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

Integral Form

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

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

where $\mathcal M$ is the Mellin transform.

For all other values of $z$ except the non-positive integers, $\Gamma \left({z}\right)$ is defined as:

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

Weierstrass Form

The Weierstrass form of the Gamma function is:

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

where $\gamma$ is the Euler-Mascheroni constant.

The Weierstrass form is valid for all $\C$.

Hankel Form

The Hankel form of the Gamma function is:

$\displaystyle \frac 1 {\Gamma \left({z}\right)} = \dfrac 1 {2 \pi i}z \oint \frac {e^t \, \mathrm d t} {t^z}$

where the contour starts at $-\infty$, circles the origin in a anticlockwise direction, and returns to $-\infty$.

The Hankel form is valid for all $\C$.

Euler Form

The Euler form of the Gamma function is:

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

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


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

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

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