Is the following statement true? Cauchy's theorem applies to any domain without singularities of f

No. f(z) = 1/z on the annulus ½ < |z| < 2 f is holomorphic there but ∮_{|z|=1} dz/z = 2πi ≠ 0 — the annulus is not simply connected.

Counterexample

Is this true?

Cauchy's theorem applies to any domain without singularities of f

No — it is false.

The counterexample

f(z) = 1/z on the annulus ½ < |z| < 2

Why it fails

f is holomorphic there but $\oint_{|z|=1}$ dz $/z = 2\pi{}i \ne 0$ — the annulus is not simply connected.

Keep practicing

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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