Applying Jordan's Lemma

Complex Analysis · Axiom Academy

EXAMPLE Applying Jordan's Lemma Evaluate a real oscillatory integral by closing a semicircular contour and summing residues. Evaluate the real integral using Jordan's lemma and the Residue Theorem. Close the real line with the upper semicircle of radius R . Jordan's lemma forces the arc's contribution to vanish as , so the whole integral equals times the residue at the one enclosed pole, z=i . Nice work — you turned a real integral with no elementary antiderivative into a single residue. The moving parts worth keeping: Trade the trig for e^ iz : , so integrate and read off the real part at the end. Sign of a picks the half-plane: for e^ iaz , the factor decays where . Here , so close upward and enclose only z=i . (If , close downward instead.) Jordan's lemma beats the plain ML bound: the oscillation lets the arc vanish even though only polynomially — the arc contributes 0 . Residue Theorem closes it: , and . The reusable pattern: for , . Our case is a=b=1 ; with a=3,b=2 it gives . Same recipe, every time: swap to e^ iaz , let the sign of a choose the half-plane, confirm Jordan kills the arc, then sum the enclosed residues.

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