Rational Function Integrals

Complex Analysis · Axiom Academy

LESSON Rational Function Integrals Turn a real integral into times the residues sitting in the upper half-plane. 1. Close the Real Line into a Loop We want , where P,Q are real polynomials. The real axis isn't a closed curve, so the residue theorem can't touch it directly. The fix: integrate over the segment [-R, R] and close it with a semicircular arc C_R sweeping through the upper half-plane. The closed loop = real segment + upper arc 2. The Arc Contributes Nothing For the loop to equal the real integral, the arc's contribution must die as . On C_R we have |z| = R , so if Q outranks P by at least two degrees, the integrand is tiny while the arc is only long. Bound times length still beats the length. If the bound is just — it need not vanish, so the method stalls. Require no real poles ( for real x ) and . Then the arc drops out and the loop integral equals the integral over the whole real line. The residue theorem says the loop integral is times the sum of residues enclosed by the contour. Our loop wraps the upper half-plane, so only poles with are inside. Their lower-half mirror images are simply ignored. For a simple pole at z_k , the residue is quick to get: . : enclosed by the loop, residue included in the sum. : outside the contour, contributes nothing. Run the recipe on the cleanest case. Check the conditions, find the single upper pole, take its residue, and multiply by — the watch below traces exactly this path, and the answer that drops out is the familiar .

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