Cauchy Theorem

If $f(z)$ is analytic in a simply connected domain, $\mathcal{D}$, and $\mathcal{C}$ is a closed curve in $\mathcal{D}$ then, ${\oint }_{\mathcal{C}}dzf(z)=0$.

${\oint }_{\mathcal{C}}dzf(z)={\oint }_{\mathcal{C}}d[x+iy][u(x,y)+iv(x,y)]={\oint }_{\mathcal{C}}(u(x,y)dx+v(x,y)dy)+i{\oint }_{\mathcal{C}}(u(x,y)dy+v(x,y)dx)={\iint }_S[\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}]dxdy+i{\iint }_S[\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}]dxdy$.