Definition:Gamma Function/Euler Form

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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 $z \in \set {0, -1, -2, \ldots}$.

Also see

Historical Note

Leonhard Paul Euler was the first to find this extension of the factorial to the real numbers.

He actually specified it in the form:

$\ds n! = \lim_{m \mathop \to \infty} \frac {m^n m!} {\paren{n + 1} \paren{n + 2} \cdots \paren{n + m}}$

He wrote to Christian Goldbach about it in a letter dated $13$th October $1729$.