\begin{definition}[Definition:Contour Integral/Complex]
Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves.
Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$.
Let $f: \Img C \to \C$ be a continuous complex function, where $\Img C$ denotes the image of $C$.
The '''contour integral of $f$ along $C$''' is defined by:
:$\ds \int_C \map f z \rd z = \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \map f {\map {\gamma_i} t} \map {\gamma_i'} t \rd t$
From Contour Integral is Well-Defined, it follows that the complex Riemann integral on the right side is defined and is independent of the parameterizations of $C_1, \ldots, C_n$.
\end{definition}