Cauchy's integral theorem can be derived from Green's theorem, as follows. Let be a simply connected open subset of , let be a holomorphic function with real and complex parts , and let be a positively oriented contour in . Then Cauchy's integral theorem states that
Note that this can be expressed in terms of two real line integrals as
Both of these integrals can be computed using Green's theorem, which gives that they are equal to
where is the interior of the region bounded by , and the integrands here both vanish by the Cauchy–Riemann equations. What this implies is that the "vector fields" (really 1-forms) and are the "gradients" (really differentials) of scalar functions, which turn out to be the real and imaginary parts of the Antiderivative of .
Cauchy's integral formula is not hard to deduce from here.
- Qiaochu Yuan https://math.stackexchange.com/users/232/qiaochu-yuan, Cauchy's integral formula and Green's theorem. Scalar or gradient?, URL (version: 2017-11-21): https://math.stackexchange.com/q/2531073