Definition:Gamma Function/Euler Form

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

• Equivalence of Definitions of Gamma Function

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: $(15)$