Applying Cauchy's Theorem

Complex Analysis · Axiom Academy

EXAMPLE Applying Cauchy's Theorem Decide whether a contour integral vanishes by locating the singularities relative to the contour. Evaluate the contour integral where the contour is the circle of radius 2 centred at the origin, traversed once counter-clockwise. The contour |z| = 2 (orange). The two singularities at 3i and −3i each sit a distance 3 from the origin, so both lie outside the contour. Nice work. You applied Cauchy's theorem the way it is meant to be used — not by computing an integral, but by checking where the integrand fails to be analytic. Analyticity is the gate: a quotient is analytic except where its denominator is zero, so the singularities are the only thing to track. Only interior singularities matter: here both poles have modulus , so they fall outside |z|=2 and the theorem applies. Result: — no parametrization or residue needed. Push the contour out to radius 4 and it would enclose both poles — the integral would no longer be 0 , and you would reach instead for the Cauchy integral formula or residues.

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