Cauchy integral theorem

Statement

Let $D\subseteq\mathbb C$ be a domain, let $f:D\to\mathbb C$ be holomorphic, and let $\gamma$ be a closed piecewise $C^1$ contour in $D$ . If $\gamma$ is null-homotopic in $D$ , then

\int_\gamma f(z)\,dz=0.

In particular, this holds for every closed contour when $D$ is simply connected.

Homological form

A stronger standard formulation replaces null-homotopy by the requirement that $\operatorname{Ind}(\gamma,a)=0$ for every $a\notin D$ . Equivalently, the cycle represented by $\gamma$ is null-homologous in $D$ . This version handles sums of contours and multiply connected domains cleanly.

Local and primitive forms

On a disc, the theorem follows from the Cauchy–Goursat theorem without assuming continuity of $f'$ . On a domain, all closed contour integrals of $f$ vanish exactly when $f$ has a holomorphic primitive. The local theorem drives the Cauchy integral formula.

References

Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw–Hill, 1979. Relevant: Chapter 4, §§1–2.