Gamma function on positive real arguments

For $a>0$ , the Gamma function is

\Gamma(a)=\int_0^\infty e^{-v}v^{a-1}\,dv.

The integral converges: $v^{a-1}$ is integrable near zero, and the exponential dominates every fixed power at infinity. In particular $\Gamma(a)>0$ .

Recurrence and smoothness

Integration by parts gives $\Gamma(a+1)=a\Gamma(a)$ , and $\Gamma(1)=1$ , so $\Gamma(n+1)=n!$ for nonnegative integers. Differentiating under the integral gives smoothness for $a>0$ ; powers of $\log v$ are integrable against a common bound when $a$ ranges over a compact positive interval. This entry uses only the positive real restriction; complex arguments require the corresponding analytic extension.

References

NIST DLMF, §5.2(i), Euler’s integral for the Gamma function.