Cauchy's integral theorem can be derived from Green's theorem, as follows. Let $ U $ be a simply connected open subset of $ \mathbb{C} $, let $ f : U \to \mathbb{C} $ be a holomorphic function with real and complex parts $ f(z) = u(z) + i v(z) $, and let $ C $ be a positively oriented contour in $ U $. Then Cauchy's integral theorem states that

$ \oint_C f(z) \, dz = \oint_C (u(z) + i v(z))(dx + i dy) = 0. $

Note that this can be expressed in terms of two real line integrals as

$ \oint_C (u \, dx - v \, dy) + i \oint_C (v \, dx + u \, dy). $

Both of these integrals can be computed using Green's theorem, which gives that they are equal to

$ \iint_D \left( - \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) \, dx \, dy + i \iint_D \left( \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} \right) \, dx \, dy $

where $ D $ is the interior of the region bounded by $ C $, and the integrands here both vanish by the Cauchy–Riemann equations. What this implies is that the "vector fields" (really 1-forms) $ u \, dx - v \, dy $ and $ v \, dx + u \, dy $ are the "gradients" (really differentials) of scalar functions, which turn out to be the real and imaginary parts of the Antiderivative of $ f $.

Cauchy's integral formula is not hard to deduce from here.

## References

- 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