PlanetPhysics/Gamma Function

The gamma function is

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

where $$x \in \mathbb{C} \setminus \{0, -1, -2, \ldots \}$$.

The Gamma function satisfies

$$ \Gamma(x+1) = x \Gamma(x) $$

Therefore, for integer values of $$x=n$$,

$$ \Gamma(n) = (n-1)! $$

Some values of the gamma function for small arguments are:

$$\begin{matrix} \Gamma(1/5)=4.5909 & \Gamma(1/4)=3.6256 \\ \Gamma(1/3)=2.6789 & \Gamma(2/5)=2.2182 \\ \Gamma(3/5)=1.4892 & \Gamma(2/3)=1.3541 \\ \Gamma(3/4)=1.2254 & \Gamma(4/5)=1.1642 \end{matrix}$$

and the ever-useful $$\Gamma(1/2)=\sqrt{\pi}$$. These values allow a quick calculation of

$$ \Gamma(n+f) $$

Where $$n$$ is a natural number and $$f$$ is any fractional value for which the Gamma function's value is known. Since $$\Gamma(x+1)=x\Gamma(x)$$, we have

$$\begin{matrix} \Gamma(n+f) & = & (n+f-1)\Gamma(n+f-1) \\ & = & (n+f-1)(n+f-2)\Gamma(n+f-2) \\ & \vdots & \\ & = & (n+f-1)(n+f-2)\cdots(f)\Gamma(f) \end{matrix}$$

Which is easy to calculate if we know $$\Gamma(f)$$.

The gamma function has a meromorphic continuation to the entire complex plane with poles at the non-positive integers. It satisfies the product formula $$ \Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^{z/n} $$

where $$\gamma$$ is Euler's constant, and the functional equation

$$ \Gamma(z) \Gamma(1-z) = \frac{\pi}{\sin \pi z}. $$

This entry is a derivative of the gamma function article from PlanetMath. Author of the orginial article: akrowne. History page of the original is here