 Gamma integral

For real $a>0$ and $\mathrm{Re}s>0$, the change of variables $u=at$ in the defining improper integral for the Gamma function givesThe improper integral converges absolutely: near zero its absolute integrand behaves like $t^{\mathrm{Re}s-1}$, and at infinity the exponential function overcomes every fixed power.^[ref]

For real $s=1,2,4$ and $a=1$, the integrands are $e^{-t}$, $t{e}^{-t}$, and $t^3{e}^{-t}$, with total areas $\Gamma (1)=1$, $\Gamma (2)=1$, and $\Gamma (4)=6$. The third integrand is taller and has six times the area of the first, although the exponential function makes each tail decay.

For real $s>0$, the kernel $t^{s-1}{e}^{-t}$ becomes the probability density function of a Gamma distribution with shape $s$ and unit rate only after division by its normalizing constant $\Gamma (s)$. In particular, the shape-four probability density function is $t^3{e}^{-t}/6$. More generally, the Gamma integral verifies that $a^s{t}^{s-1}{e}^{-at}/\Gamma (s)$ integrates to one for shape $s>0$ and rate $a>0$.

For the half-integer evaluation, the change of variables $t=u^2$ reduces the Gamma integral to the Gaussian integral:Consequently, for real $u>0$,