Applying the Residue Theorem

Complex Analysis · Axiom Academy

EXAMPLE Applying the Residue Theorem Evaluate a contour integral by finding the residues at every pole the curve encloses. Use the Residue Theorem to evaluate , where the contour is traversed once counterclockwise. Both poles of the integrand sit at distance 2 from the origin, so both lie inside the circle of radius 3. The Residue Theorem sums the residues at exactly the enclosed poles. Nice work. You turned a contour integral into a finite sum of residues — the whole point of the Residue Theorem. The theorem: , summing over only the poles z_k enclosed by C . Inside is everything: the same integrand over |z-2|=1 encloses only z=2 , giving — a different answer from the same function, just a smaller loop. Simple-pole shortcut: for f=g/h with a simple zero of h , — no factoring of h required. Higher-order poles: an order- m pole needs . For example . When the sum is zero: if f decays faster than 1/z at infinity (e.g. ), the residues at all its poles add to 0 , so a contour enclosing every pole gives 0 . Locate the poles, decide which the contour encloses, add their residues, multiply by — that sequence evaluates any such contour integral.

This is the written version of the interactive lesson above. See the full Complex Analysis course.