Fundamental Theorems of Complex Integration

Let $f$ be analytic in a simply connected domain $D$, and let $z_0$ be a fixed point in $D$. For each $z \in D$, let $C_z$ be any contour in $D$ from $z_0$ to $z$.

$F$ is analytic in $D$ and

Definite Integrals

Let $f$ be analytic in a simply connected domain $D$, and let $F$ be an antiderivative of $f$ in $D$. If $C$ is any contour in $D$ from $z_1$ to $z_2$,