TheoremProved

Cauchy's integral theorem

Statement

Let $U \subset \mathbb{C}$ be a simply connected open domain and let $f : U \to \mathbb{C}$ be a holomorphic function. Then for every piecewise smooth closed contour $\gamma$ in $U$ , the contour integral vanishes: $\oint_{\gamma} f(z)\,dz = 0$ .

Why is it true?

In multivariable calculus, a line integral around a closed loop is zero when the vector field is conservative — meaning it has no local curl or 'swirl' inside the loop. Complex differentiability is an extremely rigid condition: the Cauchy–Riemann equations force both the real and imaginary parts of $f(z)\,dz$ to be curl-free simultaneously. As long as the region has no holes where a singularity could hide and create circulation, integrating $f(z)$ along any closed path cancels out completely, and integrals between two points depend only on the endpoints.

Proof sketch

Cauchy's original 1825 approach assumes $f'$ is continuous, writes $f(z)\,dz = (u\,dx - v\,dy) + i(v\,dx + u\,dy)$ , and applies Green's theorem so the double integrals vanish by the Cauchy–Riemann equations $\partial u/\partial x = \partial v/\partial y$ and $\partial u/\partial y = -\partial v/\partial x$ . In 1900, Édouard Goursat removed the continuity assumption on $f'$ : one first proves the integral around any triangle $T \subset U$ vanishes by bisecting $T$ into four sub-triangles, picking the one with the largest integral modulus, and using the local linear approximation $f(z) = f(z_0) + f'(z_0)(z - z_0) + o(|z - z_0|)$ at the intersection point $z_0$ to show the integral is bounded by an arbitrarily small multiple of the perimeter squared. From triangles one builds a primitive on star-shaped domains and extends to general simply connected domains.

Stated by

stated 1825Augustin-Louis Cauchy

Proved by

proved 1825Augustin-Louis Cauchy

Topics that use this theorem

Functions of a complex variable

Related theorems

Cauchy's integral formula Residue theorem Stokes' theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Lars V. Ahlfors (1979). Complex Analysis
Reinhold Remmert (1991). Theory of Complex Functions