Definition:Gamma Function

The Gamma function $$\Gamma:\C \to \C \ $$ is defined, for the open right half-plane,

$$\Gamma (z) = \int_0^\infty t^{z-1} e^{-t} dt \ $$

and for all other values of $$z \ $$ except the non-positive integers as

$$\Gamma(z+1)=z\Gamma(z) \ $$.

Other equivalent definitions exists; 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 Euler's constant. The Weiestrass expression is valid for all $$\C \ $$.

Another important form of the Gamma function is the Euler form

$$\Gamma(z) = \frac{1}{z} \prod_{n=1}^\infty \left({ \left({ 1+\frac{1}{n} }\right)^z \left({1+\frac{z}{n} }\right)^{-1} }\right) \ $$,

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