Symbols:Greek/Gamma
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Gamma
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
- $\map \Gamma z$
The Gamma function $\Gamma: \C \to \C$ is defined, for the open right half-plane, as:
- $\displaystyle \map \Gamma z = \map {\MM \set {e^{-t} } } z = \int_0^{\to \infty} t^{z - 1} e^{-t} \rd t$
where $\MM$ is the Mellin transform.
For all other values of $z$ except the non-positive integers, $\map \Gamma z$ is defined as:
- $\map \Gamma {z + 1} = z \, \map \Gamma z$
The $\LaTeX$ code for \(\map \Gamma z\) is \map \Gamma z .
Euler-Mascheroni Constant
- $\gamma$
The Euler-Mascheroni constant $\gamma$ is the real number that is defined as:
| \(\displaystyle \gamma\) | \(:=\) | \(\displaystyle \lim_{n \mathop \to +\infty} \paren {\sum_{k \mathop = 1}^n \frac 1 k - \int_1^n \frac 1 x \rd x}\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \lim_{n \mathop \to +\infty} \paren {H_n - \ln n}\) |
where $H_n$ is the harmonic series and $\ln$ is the natural logarithm.
