Fundamental Theorem of Contour Integration

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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 in $D$.


Then the contour integral:

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


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:

\(\displaystyle \int_\gamma \map f z \rd z\) \(=\) \(\displaystyle \int_a^b \map f {\map \gamma t} \map {\gamma'} t \rd t\) Definition of Antiderivative
\(\displaystyle \) \(=\) \(\displaystyle \int_a^b \frac \d {\d t} \map F {\map \gamma t} \rd t\)
\(\displaystyle \) \(=\) \(\displaystyle \map F {\map \gamma b} - \map F {\map \gamma a}\) Fundamental Theorem of Calculus

$\blacksquare$