Gamma function

The factorial, generalized to non-integer inputs: $\Gamma(n+1)=n!$ for integers, defined for all real $n>0$ by an integral.

The Gamma function $\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}\,dt$ generalizes the factorial to non-integer (and complex) inputs, satisfying $\Gamma(n+1)=n!$ for nonnegative integers $n$ and $\Gamma(1/2)=\sqrt\pi$ . It appears in the denominator of the $n$ -ball volume formula for exactly this reason.

Examples