If $U$ is a [simply connected](Simply%20connected.md) domain in the complex plane, where $f$ is [holomorphic](Holomorphic%20functions.md) in $U$, and we define a closed curve $\gamma$ in $U$, then _Cauchy's theorem_ states that [closed contour integral](Closed%20contour%20integral.md) is 0: $\oint_\gamma dz f(z) =0$ Notice this theorem tells us nothing about the shape of the curve. --- # Proofs and examples ## Proof of Cauchy's theorem In order to prove it, we consider the [contour integral](Contour%20integrals.md) in much the same way we'd look at a line integral in a real [vector field](Vector%20fields.md). #MathematicalFoundations/Analysis/ComplexAnalysis/Integrals