10.1.23 Integral of a complex variable

Problem: Appropriately applying the Cauchy residue theorem, calculate the following integrals along the boundaries of infinite domains: \[ \int_{\partial \Omega} \frac{z e^{-z}}{z^{2}-1} d z, \quad \text { where } \Omega=\{\operatorname{Re} z>0\} . \]