10.1.25 Integral of a complex variable

Problem: Appropriately applying the Cauchy residue theorem, calculate the following integrals along the boundaries of infinite domains: \[ \int_{\partial \Omega} \frac{z^{3}}{(z-1)^{2}} e^{-z^{2}} d z \text {, where } \Omega=\left\{-\frac{\pi}{6}<\arg z<\frac{\pi}{6}\right\} \text {. } \]