10.1.19 Integral of a complex variable

Problem: Make sure that the multi-valued analytic functions, under the integral sign, allow the selection of a given domain \( \Omega \) of single-valued branches that satisfy the given conditions, and calculate the integral of this branch. \[ \int_{\partial \Omega} \frac{d z}{(2+\sqrt{z-1}) \sin z}, \] where \( \Omega=\left\{|z|<\frac{1}{2}\right\} \) and \( \left.\sqrt{z-1}\right|_{z=0}=i \).