Functions of a complex variable Definition

Contour integral

Definition

Let $a < b$ be real numbers and let $\gamma : [a, b] \to \mathbb{C}$ be a path that is continuous and piecewise continuously differentiable. Let $f$ be a complex function that is continuous on the image $\gamma([a, b])$ . The contour integral of $f$ along $\gamma$ is $\int_\gamma f(z) \, dz = \int_a^b f(\gamma(t)) \, \gamma'(t) \, dt.$ The integral on the right of a complex function $g$ of $t \in [a, b]$ is $\int_a^b \operatorname{Re} g(t) \, dt + i \int_a^b \operatorname{Im} g(t) \, dt$ .

Remarks

The length of $\gamma$ is $L(\gamma) = \int_a^b |\gamma'(t)| \, dt$ . The integral does not change under a reparametrization of $\gamma$ that keeps the direction. It changes sign if we reverse the direction. The path $\gamma$ is closed if $\gamma(a) = \gamma(b)$ . For a closed path, many authors write $\oint_\gamma f(z) \, dz$ . For example, along $\gamma(t) = t + i t$ with $t \in [0, 1]$ , $\int_\gamma z \, dz = \int_0^1 (1 + i) t (1 + i) \, dt = i$ .

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Complex numbers object
Contour integral definition

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Tag	`00FWNP9TXY`
URI	`m2://00FWNP9TXY`
Aliases	`curve-integral`
Kind	definition
Writing	W1 stated
MSC	30E20
Defines	Contour integral
Uses	Complex numbers

Contour integral

Definition

Remarks

Reading path

Where this leads

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