Residue

In complex analysis, a residue is a complex number that is associated with a singularity of a complex function. It is a key concept in the evaluation of complex integrals using the residue theorem.

A singularity of a function $f(z)$ is a point in the complex plane where $f(z)$ is not analytic or not defined. The most common types of singularities are isolated singularities, which are points where $f(z)$ is undefined or has a pole (a type of singularity where $f(z)$ approaches infinity as $z$ approaches the singularity).

The residue of $f(z)$ at an isolated singularity $a$ is denoted by $\text{Res}(f, a)$ and is defined as follows:

$$\text{Res}(f, a) = \frac{1}{2\pi i} \oint_C f(z) dz$$

where $C$ is a simple closed curve that encloses $a$, and is oriented counterclockwise. In other words, the residue of $f(z)$ at $a$ is equal to the value of the complex integral of $f(z)$ around a small closed curve that encloses $a$, divided by $2\pi i$.

The residue can be calculated using a number of different techniques, depending on the type of singularity. For example, if the singularity is a simple pole of order 1, the residue can be calculated using the formula:

$$\text{Res}(f, a) = \lim_{z\to a} (z-a) f(z)$$

If the singularity is a higher-order pole, the residue can be calculated using the formula:

$$\text{Res}(f, a) = \frac{1}{(n-1)!} \lim_{z\to a} \frac{d^{n-1}}{dz^{n-1}} [(z-a)^n f(z)]$$

where $n$ is the order of the pole.

In general, the residue is an important concept in complex analysis because it allows us to calculate complex integrals using the residue theorem. By computing the residues of a function at its singularities, we can determine the value of complex integrals around closed curves, which has many applications in physics, engineering, and other fields.