Symbols:Greek/Gamma
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 $\Gamma: \C \setminus \Z_{\le 0} \to \C$ is defined, for the open right half-plane, as:
- $\ds \map \Gamma z = \map {\MM \set {e^{-t} } } z = \int_0^{\to \infty} t^{z - 1} e^{-t} \rd t$
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:
| \(\ds \gamma\) | \(:=\) | \(\ds \lim_{n \mathop \to +\infty} \paren {\sum_{k \mathop = 1}^n \frac 1 k - \int_1^n \frac 1 x \rd x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to +\infty} \paren {H_n - \ln n}\) |
where $\sequence {H_n}$ are the harmonic numbers and $\ln$ is the natural logarithm.
Autocovariance
- $\gamma_k$
Let $S$ be a stochastic process giving rise to a time series $T$.
The autocovariance of $S$ at lag $k$ is defined as:
- $\gamma_k := \cov {z_t, z_{t + k} } = \expect {\paren {z_t - \mu} \paren {z_{t + k} - \mu} }$
where:
- $z_t$ is the observation at time $t$
- $\mu$ is the mean of $S$
- $\expect \cdot$ is the expectation.
Universal Gravitational Constant
- $\gamma$
Some sources use $\gamma$ to denote the universal gravitational constant:
- $\mathbf F_{a b} = \dfrac {\gamma m_a m_b {\mathbf r_{b a} } } {r^3}$
where:
- $\mathbf F_{a b}$ is the force exerted on $b$ by the gravitational force on $a$
- $\mathbf r_{b a}$ is the displacement vector from $b$ to $a$
- $r$ is the distance between $a$ and $b$
- $\gamma$ is the universal gravitational constant.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Appendix $1$: Symbols and Conventions: Greek Alphabet