I've read through Example 3.10 of Bruce Palka's text "An Introduction to Complex Function Theory" (p. 337). The example uses the residue theorem to compute $\displaystyle \int_{0}^{\infty}{\frac{dt}{t^{\lambda} (t+b)}}$ where $\displaystyle 0 < \lambda < 1$ and $\displaystyle b > 0$.

Because of the contour he chose, he uses a holomorphic branch of the log function on $\displaystyle \matbb{C} \backslash [0,\infty)$ but then proceeds to take limits and claims to compute the integral on the positive real axis, precisely where the branch that he chose is discontinuous. I'm confused; how can this be justified?