Definition:Contour Integral/Complex

Definition
Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $\gamma : \left[{ a \,.\,.\, b }\right] \to \C$ be a contour.

That is, there exists a subdivision $a_0, a_1, \ldots, a_n$ of $\left[{ a \,.\,.\, b }\right]$ such that $\gamma \restriction_{I_i}$ is a smooth path for all $i \in \left\{ {1, \ldots, n}\right\}$, where $I_i = \left[{a_{i-1} \,.\,.\, a_i}\right]$.

Here, $\gamma \restriction_{ I_i }$ denotes the restriction of $\gamma$ to $I_i$.

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

The contour integral of $f$ along $\gamma$ is defined by:


 * $\displaystyle \int_{\gamma} f \left({z}\right) \ \mathrm dz = \sum_{i \mathop = 1}^n \int_{a_{i-1}}^{a_i} f \left({\gamma \left({t}\right) }\right) \gamma' \restriction_{I_i} \left({t}\right)  \ \mathrm dt$

From Contour Integral is Well-Defined, it follows that the complex integral on the right side is defined and is independent of the choice of subdivision.

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


 * $\displaystyle \oint_{\gamma} f \left({z}\right) \ \mathrm dz = \sum_{i \mathop = 1}^n \int_{a_{i-1}}^{a_i} f \left({\gamma \left({t}\right) }\right) \gamma' \restriction_{I_i} \left({t}\right)  \ \mathrm dt$

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