Definition:Gamma Function

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Definition

Integral Form

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

$\displaystyle \map \Gamma z = \map {\mathcal M \set {e^{-t} } } z = \int_0^{\to \infty} t^{z - 1} e^{-t} \rd t$

where $\mathcal M$ is the Mellin transform.


For all other values of $z$ except the non-positive integers, $\map \Gamma z$ is defined as:

$\map \Gamma {z + 1} = z \, \map \Gamma z$


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:

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

where $\mathcal H$ is the contour starting at $-\infty$, circling the origin in an anticlockwise direction, and returning 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 $z \in \left\{{0, -1, -2, \ldots}\right\}$.


Partial Gamma Function

Let $m \in \Z_{\ge 0}$.

The partial Gamma function at $m$ is defined as:

$\displaystyle \Gamma_m \left({z}\right) := \frac {m^z m!} {z \left({z + 1}\right) \left({z + 2}\right) \cdots \left({z + m}\right)}$

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


Graph of Gamma Function

The graph of the Gamma function is illustrated here for real arguments.


The Gamma function: '"`UNIQ-MathJax-21-QINU`"' (red solid line) and '"`UNIQ-MathJax-22-QINU`"' (blue broken line)


The Gamma function:

$\Gamma \left({z}\right)$ (red solid line)
$\dfrac 1 {\Gamma \left({z}\right)}$ (blue broken line)


Also known as

Some authors call this function Euler's Gamma function, after Leonhard Paul Euler.


Examples

Gamma Function of $\dfrac 1 2$

$\map \Gamma {\dfrac 1 2} = \sqrt \pi$

Its decimal expansion starts:

$\map \Gamma {\dfrac 1 2} = 1 \cdotp 77245 \, 38509 \, 05516 \, 02729 \, 81674 \, 83341 \, 14518 \, 27975 \ldots$

This sequence is A002161 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Gamma Function of $\dfrac 1 3$

$\map \Gamma {\dfrac 1 3} = 2 \cdotp 67893 \, 85347 \, 07747 \, 63 \ldots$

This sequence is A073005 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Gamma Function of $\dfrac 1 4$

$\map \Gamma {\dfrac 1 4} = 3 \cdotp 62560 \, 99082 \, 21908 \ldots$

This sequence is A068466 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Also see

  • Results about the Gamma function can be found here.


Historical Note

The symbol $\Gamma \left({x}\right)$ for the Gamma function was introduced by Adrien-Marie Legendre.