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Evaluate a line integral

Short answer

Expand sin z / z² = 1/z − z/6 + …; simple pole at 0 with residue 1. ∮_C = 2πi if C encloses origin, 0 if not.

Evaluate ∮_C (sin z)/z² dz, where C is any closed path not passing through 0.

Let f(z) = sin z / z². Its only singularity is z = 0. Expand sin z:

sin z / z² = (z − z³/6 + z⁵/120 − ...)/z² = 1/z − z/6 + z³/120 − ...

It is a simple pole with residue = coefficient of 1/z = 1.

By Cauchy's residue theorem (C simple closed, anticlockwise):

  • If z = 0 lies inside C: ∮_C (sin z)/z² dz = 2πi × 1 = 2πi.
  • If z = 0 lies outside C: f is analytic inside and on C, so by Cauchy's theorem the integral is 0.

(Alternatively, by Cauchy's integral formula for the derivative with g(z) = sin z: ∮ g(z)/z² dz = 2πi g'(0) = 2πi cos 0 = 2πi.)

Answer: 2πi if C encloses the origin, 0 if it does not (for a closed path winding n times, 2πi·n).

✓ Answer checked by Claude Opus 5.5 (high effort)

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