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See Type 5 Integrals. I don't understand why J is equal to the original real integral multiplied by a factor of ##2\pi i##. I think the ##2\pi i## comes from the fact that as you go around ##C_2## you end up ##2\pi i## greater than when you started. But why does the difference between ##C_1, C_3## correspond to the real integral? Is it because these two segments touch the real line? I don't quite get it.