Definition:Gamma Function/Euler Form

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Definition

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


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:

$\displaystyle n! = \lim_{m \mathop \to \infty} \frac {m^n m!} {\left({n + 1}\right) \left({n + 2}\right) \cdots \left({n + m}\right)}$

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


Sources