I remember this vaguely from my university. I believe Log(z) is not defined on the whole set C. I believe there is a line going from (0, 0) under the angle "fi" (for different Log(z) functions or something) that Log(z) is not defined for points on that line. Am I correct? If true, and if your loop intersects that line, I believe that might be causing the problem. Then again, I remember that an integral is not affected if a function is not defined in a single point, but I don't know if that means anything. Is that any help?
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