Fundamental Theorem of Calculus for Contour Integrals

Theorem
Let $F, f: D \to \C$ be complex functions, where $D$ is a connected domain.

Let $C$ be a contour that is a concatenation of the directed smooth curves $C_1, \ldots, C_n$.

Let $C_k$ be parameterized by the smooth path $\gamma_k: \left[{a_k \,.\,.\, b_k}\right] \to D$ for all $k \in \left\{ {1, \ldots, n}\right\}$.

Suppose that $F$ is a primitive of $f$.

If $C$ has start point $z$ and endpoint $w$, then:


 * $\displaystyle \int_C f \left({z}\right) \, \mathrm d z = F \left({w}\right) - F \left({z}\right)$

If $C$ is a closed contour, then:


 * $\displaystyle \oint_C f \left({z}\right) \, \mathrm d z = 0$

Proof
If $C$ is a closed contour, then $z = w$.

It follows that:
 * $F \left({w}\right) - F \left({z}\right) = 0$

Also see

 * Primitive of Function on Connected Domain, for the converse of this result.