Cauchy's integral theorem

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 $f(z)=u(z)+iv(z)$, and let $C$ be a positively oriented contour in $U$. Then Cauchy's integral theorem states that

$\oint _{C}f(z)\,dz=\oint _{C}(u(z)+iv(z))(dx+idy)=0.$

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

External links

• http://users.math.msu.edu/users/shapiro/Teaching/classes/425F05/Green.pdf