Definition:Gamma Function/Integral Form

Definition

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 see

• Equivalence of Definitions of Gamma Function

• 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.

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {I}$. The Gamma function
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $16.1$: The Gamma Function: Definition of the Gamma Function
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 17.3$
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,88560 31944 \ldots$
• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $11.14$