Cauchy Theory practice

$\oint_\gamma f(z)$ dz = 0, by Cauchy's theorem.

For every a inside $\gamma, f(a) = (1/2\pi{}i) \oint_\gamma f(z)/(z - a)$ dz.

f is infinitely differentiable on D.

$\oint_\gamma f(z)$ dz = 0 even if f has a simple pole inside $\gamma$.