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:

$\ds \map \Gamma z = \map {\MM \set {e^{-t} } } z = \int_0^{\to \infty} t^{z - 1} e^{-t} \rd t$

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 $H_n$ is the harmonic series 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.

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