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 $\gamma: \closedint a b \to D$ be a contour that consists of one directed smooth curve.

Then the contour integral:

$\ds \int_\gamma \map f z \rd z = \map F {\map \gamma b} - \map F {\map \gamma a}$

Proof

By the chain rule:

$\dfrac \d {\d t} \map F {\map \gamma t} = \map {F'} {\map \gamma t} \map {\gamma'} t = \map f {\map \gamma t} \map {\gamma'} t$

Thus:

\(\ds \int_\gamma \map f z \rd z\)

\(=\)

\(\ds \int_a^b \map f {\map \gamma t} \map {\gamma'} t \rd t\)

Definition of Antiderivative

\(\ds \)

\(=\)

\(\ds \int_a^b \frac \d {\d t} \map F {\map \gamma t} \rd t\)

\(\ds \)

\(=\)

\(\ds \map F {\map \gamma b} - \map F {\map \gamma a}\)

Fundamental Theorem of Calculus

$\blacksquare$