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 $\gamma_k: \left[{a_k \,.\,.\, b_k}\right] \to D$ for all $k \in \left\{ {1, \ldots, n}\right\}$.


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


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

$\displaystyle \int_C f \left({z}\right) \rd z = F \left({w}\right) - F \left({z}\right)$

If $C$ is a closed contour, then:

$\displaystyle \oint_C f \left({z}\right) \rd z = 0$


Proof

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


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

It follows that:

$F \left({w}\right) - F \left({z}\right) = 0$

$\blacksquare$


Also see


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