# Complex Analysis/Cauchy's theorem, Cauchy's formulas and Morera's theorem

## Cauchy's theorem

Definition 5.1:

Let ${\displaystyle z_{1},z_{2},z_{3}\in \mathbb {C} }$ . Then the triangle defined by ${\displaystyle z_{1},z_{2},z_{3}}$  is defined to be the set

${\displaystyle }$.

Lemma 5.2:

Let ${\displaystyle S\subseteq \mathbb {C} }$  and ${\displaystyle f\in H(S)}$ .

Definition 5.3:

Let ${\displaystyle S\subseteq \mathbb {C} }$ . We call ${\displaystyle S}$  starshaped if and only if there exists a ${\displaystyle z^{*}\in S}$  such that

${\displaystyle \forall z\in S:[z*,z]\subseteq \mathbb {C} }$ .

Theorem 5.4: