» Required Reading » Table of Contents » Chapter 20 20.6 Integration Over Curves in the Complex Plane
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The definite integral considered so far represents area in the xy plane. We defined it however by dividing the interval in x into small subintervals (say of width d) and taking the sum of an estimate of the area of each subinterval, namely f(x')d where x' is a point in that subinterval.

Suppose now we let C be a curve in the complex plane, and let f be a function of the variable z with z = x + iy. We can define integration along that curve of an integrand f(z) by dividing the curve into small pieces, and summing f(z')((z_i-z_i-1) where z' is a point in the i-th interval whose endpoints are z_i and z_i-1, over all the pieces.
This integral along a curve will no longer represent area, since neither f nor the difference between z's will be real numbers. But we can multiply complex numbers together, so that this definition makes perfect sense.

Such an entity is called a contour integral in the complex plane. Though this integral no longer has the interpretation of area, it still has the property that it is well defined if the path C is finite and the function f is bounded and continuous on it.We denote it as follows:

Integrals of this kind are enormously valuable mathematical tools, as we may soon see.