I was exploring Cauchy's residue theorem with the gamma function and came across an interesting identity. Consider $$\int_{C_R} \Gamma(z) \, dz $$ Over the complex plane where $C_R$ is the curve defined by $Re^{i \theta}$ for $0 \leq \theta \leq 2 \pi$. From the positioning of the isolated singularities of the gamma function, it can be seen that $$\int_{C_R} \Gamma(z) \, dz = 2\pi i \sum_{n=0}^{R} \underset{z=n}{\mbox{Res}} \, \Gamma(z)$$ Now letting $R \to \infty$, so that our contour covers the entire plane, we get \begin{aligned} \lim_{R \to \infty}\int_{C_R} \Gamma(z) \, dz &= 2 \pi i \sum_{n=0}^{\infty} \underset{z=n}{\mbox{Res}} \, \Gamma(z) \\ &= 2\pi i \sum_{n=0}^{\infty} \frac{(-1)^n}{n!} \\ &=\frac{2\pi i}{e} \end{aligned}
I was wondering if this meant anything, or more generally, the significance of the sum of all the residues of any function. Does it mean anything for the contour integral over the entire plane to converge/diverge, and does the explicit value tell anything about the function?