Definition:Gamma Function

Standard Definition
The Gamma function $$\Gamma:\C \to \C \ $$ is defined, for the open right half-plane, as:
 * $$\Gamma (z) = \int_0^\infty t^{z-1} e^{-t} dt \ $$

and for all other values of $$z \ $$ except the non-positive integers as:
 * $$\Gamma(z+1)=z\Gamma(z) \ $$.

Other equivalent definitions exist, as follows.

Weierstrass Form
Of note is the Weierstrass form:
 * $$\frac{1}{\Gamma(z)} = ze^{\gamma z} \prod_{n=1}^\infty \left({ \left({ 1+\frac{z}{n} }\right) e^{\frac{-z}{n}} }\right) \ $$

where $$\gamma \ $$ is the Euler-Mascheroni constant. The Weierstrass expression is valid for all $$\C \ $$.

Euler Form
Another important form of the Gamma function is the Euler form:
 * $$\Gamma(z) = \frac{1}{z} \prod_{n=1}^\infty \left({ \left({ 1+\frac{1}{n} }\right)^z \left({1+\frac{z}{n} }\right)^{-1} }\right) = \lim_{m \to \infty} \frac {m^z m!} {z \left({z+1}\right) \left({z+2}\right) \ldots \left({z+m}\right)}$$

which is valid except for $$z \in \left\{{0, -1, -2, \ldots}\right\} \ $$.

Extension of Factorial
The Gamma function can be seen to be an extension to the complex plane of the factorial:
 * $$n! = \Gamma \left({n+1}\right) = n \Gamma \left({n}\right)$$

Hence we have:
 * $$n! = = \lim_{m \to \infty} \frac {m^n m!} {\left({n+1}\right) \left({n+2}\right) \ldots \left({n+m}\right)}$$

Historical Note
The symbol $$\Gamma \left({x}\right)$$ was introduced by Adrien-Marie Legendre.