A **contour integral** is what we get when we generalize what we’ve learned about taking integrals of real functions along a real line to integrals of complex functions along a contour in a two-dimensional complex plane.

A *contour *is a loop around the negative x-axis:

It’s not quite as difficult as it sounds. To directly calculate the values of a contour integral around a given contour, all we need to do is **sum the values of the “complex residues**“, inside of the contour. A residue in this case is what remains when you integrate around the origin. We can also apply the * Cauchy integral formula*, or use an application of the residue theorem.

**What is a contour in the complex plane?** Think about it as a finite (fixed) number of smooth curves. We can define it more exactly as a directed curve, that is made up of a finite sequence of directed smooth curves. Each of these curves must be matched to give just one direction.

## Integrating a Contour Integral

Integrating over a contour might sound intimidating, so let’s start with something a bit simpler. Suppose we want to integrate the function f(x) over the curve Γ, and suppose M ∈ ℂ^{1}[I] defines a curve such that Γ = M(I).

Then we can define contour integration over our curve Γ as

That’s all well and good. But what if we want to integrate over a contour which is defined by M_{1},…M_{l} ∈ C^{1}[I]? We could describe our contour this way:

It turns out all that we need to do is define integration piecewise, and everything we’ve already learned carries over.

## References

Chong, Y. D. MH2801 Course Notes: Complex Methods for the Sciences. Section 8, Contour Integration. Published 2016. Retrieved from

http://www1.spms.ntu.edu.sg/~ydchong/teaching/08_contour_integration.pdf on August 27, 2019

Olver, Sheehan. Class Notes: Lesson 8: Contour Integration Department of Mathematics, Imperial College. Retrieved from http://www.maths.usyd.edu.au/u/olver/teaching/NCA/08.pdf on August 27, 2019