Applying the ML Inequality

Complex Analysis · Axiom Academy

EXAMPLE Applying the ML Inequality Bounding a complex contour integral with the estimation lemma, then letting the radius grow Let C_R be the upper semicircle , , of radius R > 1 centered at the origin. Show that , and conclude that this integral as . Nicely done — you bounded a contour integral without ever evaluating it, and showed the arc's contribution dies off as . The ML inequality: , where L is the arc length of C and M is any upper bound for |f(z)| on C . You never integrate — you just bound and measure. Bound top and bottom separately: on the arc (since ), and the reverse triangle inequality gives , so . The payoff: as — the numerator grows like R , the denominator like R^2 . This "vanishing arc" argument is the engine behind evaluating real integrals by residues: kill the semicircle's contribution, and only the residues inside survive.

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