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gamma function (Definition)

The gamma function is

$\displaystyle \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

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

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

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

Some values of the gamma function for small arguments are:

\begin{displaymath}\begin{array}{cc} \Gamma(1/5)=4.5909 & \Gamma(1/4)=3.6256 \\ ... ...=1.3541 \ \Gamma(3/4)=1.2254 & \Gamma(4/5)=1.1642 \end{array}\end{displaymath}

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

$\displaystyle \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


$\displaystyle \Gamma(n+f)$ $\displaystyle =$ $\displaystyle (n+f-1)\Gamma(n+f-1)$  
  $\displaystyle =$ $\displaystyle (n+f-1)(n+f-2)\Gamma(n+f-2)$  
  $\displaystyle \vdots$    
  $\displaystyle =$ $\displaystyle (n+f-1)(n+f-2)\cdots(f)\Gamma(f)$  

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

$\displaystyle \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

$\displaystyle \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



"gamma function" is owned by bloftin.

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Cross-references: Euler's constant, formula, function's
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This is version 1 of gamma function, born on 2006-05-07.
Object id is 167, canonical name is GammaFunction.
Accessed 1326 times total.

Classification:
Physics Classification: 02.30.Gp (Special functions)
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