Residue theorem: ∮(1-cos z)/z³ dz

Here C is taken as any simple closed contour enclosing z = 0 (for example |z| = 1), since the only singularity of f is at the origin.

f(z) = (1 − cos z)/z³

Nature of the singularity: the numerator 1 − cos z = z²/2 − z⁴/24 + … has a zero of order 2 at z = 0, the denominator has a zero of order 3. So f(z) = 1/(2z) − z/24 + … and z = 0 is a simple pole.

Residue:

Res at 0 = lim (z → 0) z f(z) = lim (1 − cos z)/z² = 1/2

(Laurent series: f(z) = 1/(2z) − z/24 + z³/720 − …; coefficient of 1/z is 1/2.)

By the residue theorem:

Answer: ∮_C (1 − cos z)/z³ dz = 2πi × 1/2 = πi