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, \(\oint_\mathcal{C} dz f(z) = 0\).

\(\oint_\mathcal{C}dz f(z) = \oint_\mathcal{C}d[x+iy] [u(x,y) + iv(x,y)] = \oint_\mathcal{C}(u(x,y)dx + v(x,y)dy) + i\oint_\mathcal{C}(u(x,y)dy + v(x,y)dx) = \iint_S\left[\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\right]dxdy + i\iint_S\left[\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}\right]dxdy\).