Example 2

Consider the integral $\int_{0}^{+\infty} t^{\lambda-1} e^{-t} \, dt$. Since $e^{-t} \sim 1$ on a neighbourhood of 0, one must have $\lambda > 0$ if this integral is to converge; this condition is also sufficient since on a neighbourhood of $+\infty$ one has $e^{-t} \ll t^{-\mu}$ for any $\mu > 0$. When $x > 0$ the integral is called the Eulerian integral of the second kind and is denoted by $\Gamma(x)$ (cf. VII, p. 311).