ResidueTheorem

Let $C$ be a positively oriented simple closed curve in the complex plane that encloses a region $D$. Suppose that $f(z)$ is a complex function that is analytic in $D$ except for a finite number of isolated singularities. Then, the value of the complex integral of $f(z)$ around $C$ is given by:

$$\oint_C f(z) dz = 2\pi i \sum Res(f, a)$$

where the sum is taken over all the singularities $a$ of $f(z)$ that lie inside $C$, and $Res(f, a)$ denotes the residue of $f(z)$ at the singularity $a$.

In other words, the residue theorem tells us that the value of a complex integral around a closed curve is determined by the singularities of the function inside the curve. The residues of the function at those singularities can be computed using techniques such as Laurent series expansions and then added up to obtain the value of the integral.