Complex Analysis & Residues

Contour integration and the Residue Theorem

∮_C f(z) dz = 2πi · Σ Res(f, zₖ)
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The Residue Theorem states the contour integral depends only on poles inside C. Poles appear as color singularities (domain coloring: hue = argument, brightness = magnitude). The integral equals 2πi times the sum of residues at enclosed poles. Branch cuts are shown as discontinuities. Click to toggle poles inside/outside the contour.