Definition:Contour Integral/Complex

Definition
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 $\gamma_i: \left[{a_i\,.\,.\,b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$.

Let $f: \operatorname{Im} \left({C}\right) \to \C$ be a continuous complex function, where $\operatorname{Im} \left({C}\right)$ denotes the image of $C$.

The contour integral of $f$ along $C$ is defined by:


 * $\displaystyle \int_C f \left({z}\right) \ \mathrm dz = \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right)  \ \mathrm dt$

From Contour Integral is Well-Defined, it follows that the complex integral on the right side is defined and is independent of the parameterizations of $C_1, \ldots, C_n$.

Contour Integral along Closed Contour
If $C$ is a closed contour, we use the symbol $\displaystyle \oint$ for the contour integral, but the definition remains the same:


 * $\displaystyle \oint_C f \left({z}\right) \ \mathrm dz = \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right)  \ \mathrm dt$

Also known as
A contour integral is called a line integral or a curve integral in many texts.

Also denoted as
There is no standard in literature for the use of the symbols $\displaystyle \int$ and $\displaystyle \oint$.