Complex Integral
A complex integral is an integral over a complex line segment. Compared to a normal integral from \(a\) to \(b\), where there is only one path from start to end, on the complex plane, there are infinite ways to connect two points. Thus, the integration must be defined a bit differently.
Line Integral¶
The complex line integral (also path- or curve integral) from a point \(a\) to a point \(b\) must define exactly which line through the complex to integrate along. We define this line by defining a parametrization function \(\varphi(x)\) that projects real values to the complex values \(z\) on the complex line \(L\). We then integrate over \(f(z)=f(\varphi(x))\) with \(dz=\varphi^\prime(x)\):
Example¶
Given \(f(z)=\frac{1}{z}\) over the unit circle \(\varphi(x)=e^{jx}\):
we can use \(f(\varphi(x))=\frac{1}{e^{jx}}\) and \(\varphi^\prime(x)=je^{jx}\) to solve the integral: