I thought that a justification of inversing the limit-integral in a discussion (moment 5:50) of the integral of the sinc function using the Euler Integral was needed, it inversed the limit integral :
$$ \lim_{s\rightarrow0} \int_0^{\infty} \{ u^{s-1} \sin(u) \} du \qquad \text{getting} \quad \int_0^{\infty} \{ u^{-1} \sin(u) \} du$$
Under my current understanding, if we take $f(s,u) = u^{s-1} sin(u)$ and $\lim_{s\rightarrow 0} f(s,u) = \phi(u) = u^{-1} \sin(u)$ , we need to apply something like dominating it for example $\|f(s,u) \| \leq g(u) $ by an integrable function on it's domain of integration so that we get $ \lim_{s\rightarrow 0}F(s) = \int \phi(u)du$
Is such an application of some result is needed here indeed or did I miss something?