Definition:Gamma Function/Euler Form
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Definition
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 $1729$.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $16.11$: Other Definitions of the Gamma Function
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 17.6$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: $(15)$