We know that for $Re(z) > 0$ we have per definition$$\Gamma(z) = \int_{0}^{\infty}e^{-t}t^{z-1}dt$$ and this is well defined for the domain. If we extend the function into a unique meromorph function on $\mathbb{C}$, are we still allowed to use the definition above? I read somewhere that the problem with the formula is that the integrand is not absolute integrable near $0$. What does this mean?
Another alternative formula would be
$$\Gamma(z) = \sum^{\infty}_{n=0}\frac{(-1)^n}{n!}\frac{1}{z+n}+\int^{\infty}_{1}e^{-t}t^{z-1} dt$$.
How is this formula not problematic compared to the previous one when we consider the domain to be the entire $\mathbb{C}$?