Definition:Gamma Function/Integral Form

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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:

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

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.

Also see

  • Results about the gamma function can be found here.

Historical Note

The integral form of the gamma function $\Gamma \left({z}\right)$ was discovered by Leonhard Paul Euler.