10.1.20 Integral of a complex variable

Problem: Make sure that the multi-valued analytical functions, standing under the sign of the integral, allow selection of the given domain \( \Omega \) of singlevalued branches that satisfy the given conditions, and calculate the integral of this branch. \[ \int_{\partial \Omega} \frac{d z}{\ln z-3 \pi i}, \] where \( \Omega=\left\{|z+2|<\frac{3}{2}\right\} \) and \( \left.\ln z\right|_{z=-e}=1-\pi i \).