# Poles, residues and contour integrals

> The coefficient of (z−a)^(−1) in the Laurent expansion is the residue.

Course: [Advanced mathematical methods](https://studybrix.com/en/courses/advanced-mathematical-methods/index.md)
Level: advanced
Prerequisites: Complex differentiability and Laurent series.

## Notes

### Residue

The coefficient of (z−a)^(−1) in the Laurent expansion is the residue.

### Simple pole

For p(z)/(z−a), with analytic p, the residue is p(a).

### Contour theorem

A positively oriented simple closed contour integrates a meromorphic function to 2πi times enclosed residues, with no poles on the contour.

### Orientation

Reversing contour direction reverses the sign of the integral.

## Practice questions

### 1. Find the residue of (z²+1)/(z−2) at z=2.

Answer: 5

Explanation: Evaluate the analytic numerator at 2: 4+1=5.

### 2. Find the residue of z²/(z−1)² at z=1.

Answer: 2

Explanation: For the double pole, differentiate the numerator and evaluate: 2z at 1 gives 2.

### 3. Which Laurent coefficient is the residue at a?

A. The constant coefficient
B. Always the highest positive power
C. The coefficient of (z−a)^(−1)
D. The coefficient of (z−a)^(−2)

Answer: The coefficient of (z−a)^(−1)

Explanation: This is the definition of a residue.

### 4. Integrate 1/z counterclockwise around the unit circle. Enter the coefficient of 2πi.

Answer: 1

Explanation: The only enclosed pole has residue 1.

### 5. Why does the standard residue formula need modification or a different setup when a pole lies on the contour?

Sample answer / marking guidance: The ordinary contour integral is not defined through a singularity in the usual way. One must specify an indentation, principal value or another limiting procedure before applying suitable results.

Explanation: Credit the undefined ordinary path integral and the need for a declared treatment.

### 6. Integrate 3/z clockwise around the unit circle. Enter the coefficient of 2πi.

Answer: -3

Explanation: The residue is 3 and clockwise orientation changes the sign.

### 7. Find the residue of 1/[z(z−1)] at z=0 and explain the limit.

Sample answer / marking guidance: It is −1, because zf(z)=1/(z−1), which tends to −1.

Explanation: Credit cancellation of the simple pole factor and evaluation.

### 8. If a function is holomorphic inside and on a closed contour in a simply connected region, what is its contour integral?

A. Always 2πi
B. 0
C. Its value at the origin
D. Undefined

Answer: 0

Explanation: Cauchy's integral theorem applies under these conditions.

## Further reading

- https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
- https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.