The Gamma Function

Definition of the gamma function as an improper integral, its values at positive integers (generalizing the factorial), the recursion $\Gamma(x+1)=x\Gamma(x)$ , and evaluation at positive half-integers using $\Gamma(1/2)=\sqrt{\pi}$ .

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Tutorial

Definition of the Gamma Function

The gamma function is a continuous extension of the factorial. For $x>0$ , it is defined by the improper integral

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

Its most important property is that, for any positive integer $n,$

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

Notice the shift by one: the argument of $\Gamma$ is one larger than the number whose factorial we are taking. For example,

\Gamma(1)=0!=1,\quad \Gamma(2)=1!=1,\quad \Gamma(3)=2!=2,\quad \Gamma(4)=3!=6.