Laurent

calculus over \(\mathbb{C}\) is not a cosmetic upgrade of real analysis — it is a different subject with better theorems. asking a function of a complex variable to be differentiable once forces it to be differentiable infinitely often, equal to its taylor series, and rigid enough that its values on a tiny arc determine it everywhere. the payoff for this rigidity: integrals that real methods cannot touch fall to a residue computation in three lines (Brown, James W. and Churchill, Ruel V., 2009).