Contours and Contour Integrals A contour in the complex plane is a closed path in the complex plane that can be traced by a real variable. The integral of a...
Contours and Contour Integrals
A contour in the complex plane is a closed path in the complex plane that can be traced by a real variable. The integral of a function over a contour is a measure of the total area enclosed by the contour.
A contour integral can be interpreted as the sum of the values of the function evaluated at each point in the contour.
Contour integrals are used in complex analysis to evaluate integrals over closed curves. Cauchy's Theorem states that the value of an integral over a closed contour is equal to the area of the region enclosed by the contour.
Examples:
The contour integral of f(z) = z^2 over the unit circle C is pi.
The contour integral of f(z) = 1/z over the real axis is infinity.
The contour integral of f(z) = e^z over the imaginary axis is 0