Residue Definition

Complex Analysis · Axiom Academy

One coefficient of the Laurent series controls an entire contour integral — and it has a name. 1. The Residue Is a Single Coefficient Let f have an isolated singularity at z_0 , so near z_0 it has a Laurent expansion in powers of (z-z_0) . The residue of f at z_0 is the coefficient of the (z-z_0)^ -1 term — written a_ -1 . the residue is just one of its coefficients 2. Why Only That Coefficient Survives Integrate the Laurent series term by term around a small circle enclosing z_0 . Each power (z-z_0)^n gets its own integral — and almost all of them vanish. As you go once around the loop, every term comes back to where it started, so its integral is zero . Only the n=-1 term keeps adding the same value all the way around, accumulating to . . The running value traces a closed loop and returns to the origin — net contribution zero. . The integrand is constant around the loop, so the value climbs steadily to . Multiply through by a_ -1 and sum: every term contributes 0 except the -1 term, which contributes . Dividing by recovers exactly the residue — which is why the contour integral has its own closed form. 3. Residues Add Up: The Residue Theorem Now put several singularities inside one contour. Because integration is linear and each singularity contributes only its own a_ -1 , the whole contour integral is just times the sum of the residues enclosed. This is the payoff: a potentially brutal integral becomes a short list of coefficients added together.

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