The residue of a pole of a function is an important concept in complex analysis, specifically in the context of evaluating integrals using the Residue Theorem. To find the residue of a function at a pole, you need to know the type of pole (whether it is a simple pole, a higher-order pole, or an essential singularity) and apply the appropriate method for calculating the residue.

1. Simple Pole (Pole of order 1)

If $z_0$ is a simple pole of a function $f(z)$, the residue at $z_0$ is given by:

$\text{Res}(f, z_0) = \lim_{z \to z_0} (z - z_0) f(z)$

This formula essentially isolates the coefficient of the $\frac{1}{z - z_0}$ term in the Laurent series of $f(z)$ about $z_0$.

Example:

Suppose $f(z) = \frac{1}{z(z - 2)}$. To find the residue at $z = 2$ (a simple pole): $\text{Res}(f, 2) = \lim_{z \to 2} (z - 2) \frac{1}{z(z - 2)} = \frac{1}{2}$

2. Pole of Higher Order (Pole of order $n$)

If $z_0$ is a pole of order $n$, the residue is calculated using the following formula:

$\text{Res}(f, z_0) = \frac{1}{(n-1)!} \lim_{z \to z_0} \frac{d^{n-1}}{dz^{n-1}} \left[(z - z_0)^n f(z)\right]$

This formula involves differentiating the function multiple times, depending on the order of the pole.

Example (Pole of order 2):

Let $f(z) = \frac{1}{(z - 1)^2 (z - 2)}$. At $z = 1$, we have a second-order pole: $\text{Res}(f, 1) = \lim_{z \to 1} \frac{d}{dz} \left[ (z - 1)^2 f(z) \right] = \lim_{z \to 1} \frac{d}{dz} \frac{1}{z - 2} = \frac{1}{(1 - 2)^2} = -1$

3. Essential Singularities

If $z_0$ is an essential singularity, the function does not have a residue in the standard sense, and other methods (such as calculating the Laurent series) are needed to analyze the singularity.

General Steps to Find Residue:

1. Identify the pole $z_0$ and its order.
2. If it's a simple pole, apply the simple residue formula.
3. For higher-order poles, use the differentiation formula.
4. Simplify the result to obtain the residue.

Would you like more details or clarifications on any of these methods?

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Here are some related questions:

1. How do you find the residue for a rational function?
2. What is the geometric interpretation of a residue?
3. How does the residue theorem help in evaluating integrals?
4. How do you handle essential singularities in complex functions?
5. Can residues be used in real analysis for solving integrals?

Tip: In complex analysis, poles are points where a function goes to infinity. The residue isolates the behavior of a function near such points and is a key to evaluating contour integrals.