Fundamental Theorem of Calculus for Contour Integrals/Corollary

Theorem
Let $D \subseteq \C$ be an open set.

Let $f: D \to \C$ be a continuous function.

Suppose that $F: D \to \C$ is an antiderivative of $f$.

Let $\gamma: \left[{a \,.\,.\, b}\right] \to D$ be a contour in $D$.

Then the contour integral:
 * $\displaystyle \int_\gamma f \left({z}\right) \ \mathrm d z = F \left({\gamma \left({b}\right)}\right) - F \left({\gamma \left({a}\right)}\right)$

Proof
By the chain rule:
 * $\dfrac {\mathrm d} {\mathrm d t} F \left({\gamma \left({t}\right)}\right) = F' \left({\gamma \left({t}\right)}\right) \gamma' \left({t}\right) = f \left({\gamma \left({t}\right)}\right) \gamma' \left({t}\right)$

Thus: