Very helpful, thank you! Bounding the integral on the upper arc of the semicircle to show that it goes to 0 in the limit takes some extra work, though.
What about the convergence or real integral in the disk where we convert and integrate? If real integral doesn't convergence in the disk, it's impossible to do next steps.
This is way late but answering is helping me understand. The difference is that the upper arc of the semicircle contributes to the value of the integral. You are correct that the actual contour integral in the complex plane has the same value for both sets of bounds, but in the infinite case the value of the integral is entirely from the real line, as the upper arc contributes 0. The Wikipedia article on Residue Theorem was helpful for understanding the semicircle case.
