My book on complex analysis dedicates an entire chapter to this integral in order to motivate/define complex line integrals. Having spent a semester on integration theory, I am not to keen on learning about yet another integral when I am perfectly satisfied with the Lebesgue integral. My understanding is that the Riemann-Stieltjes integral is just a generalization of the Riemann integral. I simply don't see why I need it for this course or why I would ever need it at all. Some insight would be greatly appreciated.
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I can give three reasons: