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.

Learn how to find the residue of a pole in complex analysis, covering concepts like simple and higher-order poles, essential singularities, and the Residue Theorem. Includes formulas and examples for different types of poles. Suitable for advanced mathematics students.

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