Cauchy-Goursat Theorem

Theorem
Let $$U \ $$ be a simply connected open subset of the complex plane $$\mathbb{C} \ $$, and $$a_1, a_2, \dots, a_n \ $$ finitely many points of $$U \ $$.

Let $$f:U \to \mathbb{C} \ $$ be analytic in $$U - \left\{{a_1, a_2, \dots, a_n }\right\} \ $$.

If $$L = \partial U \ $$ oriented counterclockwise, then

$$\oint_L f(z) dz = 2\pi i \sum_{k=1}^n \text{ Res}( a_k ) \ $$

where $$\text{ Res} \ $$ is the residue of $$ f \ $$ at a point.