Is the following statement true? A small indentation around a simple pole contributes 2πi·Res

No. The indentation at 0 when computing ∫₀^∞ sin x/x dx A half-circle contributes iπ·Res; using 2πi gives twice the right answer.

Counterexample

Is this true?

A small indentation around a simple pole contributes $2\pi{}i\cdot$ Res

No — it is false.

The counterexample

The indentation at 0 when computing $\int_{0}^\infty \sin x/x$ dx

Why it fails

A half-circle contributes $i\pi\cdot$ Res; using $2\pi{}i$ gives twice the right answer.

Keep practicing

Execution slip

The idea was right. The computation was not.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.