Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. For which values of the real parameter $\alpha$ is the function $x \mapsto x ^ { \alpha } W ( x )$ integrable on $[ 1 , + \infty [$?
Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. For which values of the real parameter $\alpha$ is the function $x \mapsto x ^ { \alpha } W ( x )$ integrable on $[ 1 , + \infty [$?