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My difficulty is this: for the point to be considered a branch point, f(r,θ) and f(r,θ+2π) must be different for ANY closed path. If the branch point is zero, you can arbitrarily shrink the path and still be sweeping from 0 to 2π. If your branch point is anything but the origin, then as the path becomes arbitrarily small, so does the angle you sweep over. In such a case, how can you get multiple values for the same input?

Thanks in advance for any answers to this problem.