Consider the integral $$I(m,n):=\int_0^1 \ln^m(x)x^n dx$$
With the definition of the gamma-function, I worked out that
$$I(m,n)=\frac{(-1)^mm!}{(n+1)^{m+1}}$$
for $m,n\ge 0$. With the definitions $m!:=\Gamma(m+1)$ and $(-1)^m:=e^{i\pi m}$, the formula seems to be correct even for real non-negative $m,n$.
Questions :
Is the formula actually correct for all non-negative real $m,n$ ?
Can the formula be extended for complex numbers $m,n$ , assuming $\Gamma(m+1)$ is defined and $n\ne -1$ ? If not, for which complex values $m,n$ does the given formula hold and for which complex $m,n$ does the integral exist ?