hi, im trying to understand the conformal map $f$ in this image
imgur.com/a/O1N4JIy , it maps the upper half plane to some polygon. I understand the authors argument saying that $\arg(f')$ jumps at the $z_k$ , but he later then says that since $f_k = (z-z_k)^{- \beta_k}$ jumps the same amount at $z_k$, $\frac{f'}{f_k}$ is holomorphic in a neighbourhood of $z_k$, this would make sense to me if we knew $|f'|$ did not jump at $z_k$, but im not sure how to see this