10.1.21 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 single-valued branches that satisfy the given conditions, and calculate the integral of this branch. \[ \int_{\partial \Omega} \frac{z+2}{2 \pi i-\ln (1+z)}, \] where \( \Omega=\left\{|z-2|<\frac{5}{2}\right\} \) and \( \left.\ln (1+z)\right|_{z=e-1}=1-2 \pi i \).