Definition:Gamma Function
Definition
Integral Form
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:
- $\map \Gamma {z + 1} = z \map \Gamma z$
Weierstrass Form
The Weierstrass form of the gamma function is:
- $\ds \frac 1 {\map \Gamma z} = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac z n} e^{-z / n} }$
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:
- $\ds \frac 1 {\map \Gamma z} = \dfrac 1 {2 \pi i} \oint_\HH \frac {e^t \rd t} {t^z}$
where $\HH$ 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:
- $\ds \map \Gamma z = \frac 1 z \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac 1 n}^z \paren {1 + \frac z n}^{-1} } = \lim_{m \mathop \to \infty} \frac {m^z m!} {z \paren {z + 1} \paren {z + 2} \cdots \paren {z + m} }$
which is valid except for $z \in \set {0, -1, -2, \ldots}$.
Partial Gamma Function
Let $m \in \Z_{\ge 0}$.
The partial gamma function at $m$ is defined as:
- $\ds \map {\Gamma_m} z := \frac {m^z m!} {z \paren {z + 1} \paren {z + 2} \cdots \paren {z + m} }$
which is valid except for $z \in \set {0, -1, -2, \ldots, -m}$.
Graph of Gamma Function
The graph of the gamma function is illustrated here for real arguments.
The gamma function:
- $\map \Gamma z$ (red solid line)
- $\dfrac 1 {\map \Gamma z}$ (blue broken line)
Also known as
Some authors refer to the gamma function as Euler's gamma function, after Leonhard Paul Euler.
Some French sources call it the Eulerian function.
Examples
Gamma Function of $4$
- $\map \Gamma 4 = 6$
Gamma Function of $\dfrac 1 2$
- $\map \Gamma {\dfrac 1 2} = \sqrt \pi$
Gamma Function of $\dfrac 1 3$
- $\map \Gamma {\dfrac 1 3} = 2 \cdotp 67893 \, 85347 \, 07747 \, 63 \ldots$
Gamma Function of $\dfrac 1 4$
- $\map \Gamma {\dfrac 1 4} = 3 \cdotp 62560 \, 99082 \, 21908 \ldots$
Also see
- Zeroes of Gamma Function
- Poles of Gamma Function
- Gamma Function Extends Factorial
- Gamma Difference Equation
- Results about the gamma function can be found here.
Historical Note
The symbol $\map \Gamma z$ for the gamma function was introduced by Adrien-Marie Legendre.