Cauchy’s Theorem

Theorem (Cauchy’s Theorem (Homotopy Version))

Let $G$ be a connected open subset of the complex plane. Let $f:G\to \mathbb{C}$ be a holomorphic function. If ${\gamma }_0$, ${\gamma }_1$ are homotopic closed curves in $G$, then

$${\int }_{{\gamma }_0}f(z)dz={\int }_{{\gamma }_1}f(z)dz$$

If $\gamma$ is a null-homotopic closed curve in $G$, then

$${\int }_{\gamma }f(z)dz=0$$

Cauchy’s Theorem has a converse:

Morera’s Theorem

Let $G\subset \mathbb{C}$ be open and let $f:G\to \mathbb{C}$ be a continuous function. If ${\int }_{\gamma }f(z)dz=0$ for every contour $\gamma$ contained in $G$, then $f$ is holomorphic in $G$.