The Gamma function \(\Gamma(z) = \int_0^\infty x^{z-1} e^{-x}\,\mathrm{d}x\) is originally defined only for \(t > 0\); it can however be analytically continued to negative values of \(z\). Show that, as \(z \to -n\), where \(n \in \mathbb{N}_0\), \(\Gamma(z)\) has poles; compute the order and the residue of these poles.
Erratum: the source’s condition “\(t > 0\)” involves a variable that does not occur in the problem. The defining integral converges exactly for \(\operatorname{Re} z > 0\): at \(z = 0\) the integrand behaves like \(x^{-1}\) near \(x = 0\) and the integral diverges. The intended condition is \(\operatorname{Re} z > 0\).