I believe I figured it out. We fix a sheet of and we integrate on that sheet. The sheet has a cut from to and therefore the integral will be defined up to an integral number of cycles around the cut. But the integral along a simple closed curve whose interior contains the cut is equal to the integral along the cut itself, by Cauchy's theorem (since whatever branch of which we have is analytic in the region between the two curves). The integral along the cut is , which is indeed a generator for the group of periods of .