A while ago I asked a question (link at the bottom) about the gamma function. It was answered with:
'$\int\limits_a^b e^{-t} t^{x-1}dt=\int\limits_a^b |e^{-t} t^{z-1}|dt$ with $x\in\mathbb{R}_{>0}$ and $z\in\mathbb{C}_{\Re(z)>0}$. Since the LHS converges, we have that $\int\limits_a^b e^t t^{z-1} dt$ converges.'
Why can we draw this conclusion? In my mind we can only draw conclusions about $\int|f(z)|dt$ and not about $\int f(z) dt$.
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Link: Does convergence of $\Gamma(x)$ imply convergence of $\Gamma(z)$? Is it generalisable?