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 $\gamma_k: \closedint {a_k} {b_k} \to D$ for all $k \in \set {1, \ldots, n}$.

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

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


 * $\displaystyle \int_C \map f z \rd z = \map F w - \map F z$

If $C$ is a closed contour, then:


 * $\displaystyle \oint_C \map f z \rd z = 0$

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

It follows that:
 * $\map F w - \map F z = 0$

Also see

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