Fundamental Theorem of Calculus for Contour Integrals

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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 an antiderivative of $f$.


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

$\ds \int_C \map f z \rd z = \map F w - \map F z$


If $C$ is a closed contour, then:

$\ds \oint_C \map f z \rd z = 0$


Corollary

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

\(\ds \int_C \map f z\) \(=\) \(\ds \sum_{k \mathop = 1}^n \int_{a_k}^{b_k} \map f {\map {\gamma_k} t} \map {\gamma_k'} t \rd t\) Definition of Complex Contour Integral
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \int_{a_k}^{b_k} \paren {\dfrac \rd {\rd t} \map F {\map {\gamma_k} t} } \rd t\) Derivative of Complex Composite Function
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \paren {\map F {\map {\gamma_k} {b_k} } - \map F {\map {\gamma_k} {a_k} } }\) Fundamental Theorem of Calculus for Complex Riemann Integrals
\(\ds \) \(=\) \(\ds \map F {\map {\gamma_n} {b_n} } - \map F {\map {\gamma_1} {a_1} }\) the sum is telescoping
\(\ds \) \(=\) \(\ds \map F w - \map F z\) Definition of Endpoints of Contour (Complex Plane)


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

It follows that:

$\map F w - \map F z = 0$

$\blacksquare$


Also see


Sources

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