Trigonometric Integrals

Complex Analysis · Axiom Academy

LESSON Trigonometric Integrals Wrap the interval into the unit circle and let residues do the integral. 1. Wrap the Interval into a Circle Set . As the angle runs along the real interval , the point z travels once counterclockwise around the unit circle |z|=1 — starting and ending at z=1 . The one-dimensional interval has become a closed loop in the complex plane. On the unit circle and . Adding and subtracting these recovers the trig functions as the point's shadows — its real part is , its imaginary part is . The differential converts too, from . — the horizontal projection of z . — the vertical projection of z . Any rational becomes a rational function of z . The Residue Theorem closes the deal: the contour integral equals times the sum of residues at the poles inside the circle. Poles outside are simply ignored. Watch it on the workhorse integral with a>|b| : The same canonical answer holds for . A trigonometric integral over a full period is a contour integral in disguise — substitute , rewrite the trig functions and in z , and read off the answer from the residues inside the unit circle. Scroll up to revisit any step.

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