Cauchy-Goursat

definition (Complex Integral):

$\begin{align*} \int_{\gamma} f d z:=\int_{I} f(\gamma(t)) \gamma^{\prime}(t) \,dt = \int_\gamma (u+iv)\,dx+ (-v+iu)\,dy .\end{align*}$

theorem (Cauchy-Goursat Theorem):

If $f$ is holomorphic on a region $\Omega$ with $\pi_1 \Omega = 1$ , then for any closed path $\gamma \subseteq \Omega$ , $\begin{align*} \int_{\gamma} f(z) \,dz= 0 .\end{align*}$

slogan:

Closed path integrals of holomorphic functions vanish.

proof (of Cauchy):

Apply Stokes’: $\begin{align*} \oint_{\partial D} f(z) d z=\int_{D} d(f(z) d z)=\int_{D}\left(\frac{\partial f}{\partial z} d z+\frac{\partial f}{\partial \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu} d \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu\right) \wedge d z=\int_{D} \frac{\partial f}{\partial z} d z \wedge d z+0 d \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu \wedge d z=0 .\end{align*}$