---
title: "Residues of a function"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:residues-of-a-function-four"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/residues-of-a-function-four
site: BCT Engineering
---

# Residues of a function

Mathematics III · Chapter 2: Line, Surface and Volume Integrals

- Asked in: 2081 Chaitra (new course), 2 marks
- Video: [How to find the Residues of a Complex Function](https://www.youtube.com/watch?v=sSj7z-pz-yY) — Faculty of Khan
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/residues-of-a-function-four
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Find the residue of f(z) = (3z − 4)/((z − 2)(z − 3)) at each of the singularities.

## Short answer

Simple poles at z=2 and z=3. Residues: at z=2, residue = −2; at z=3, residue = 5.

## Answer

f(z) = (3z − 4)/((z − 2)(z − 3)) has simple poles at z = 2 and z = 3.

```text
At z = 2:  Res = (3z − 4)/(z − 3) at z = 2 = 2/(−1) = −2
At z = 3:  Res = (3z − 4)/(z − 2) at z = 3 = 5/1 = 5
```

**Answer:** residue = −2 at z = 2 and 5 at z = 3. (Check: f = −2/(z − 2) + 5/(z − 3); sum of residues = 3 = lim z f(z).)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Simple poles at:
   A. z=-2,-3
   B. z=0
   C. z=4/3
   D. z=2 and z=3
   Answer: D. Roots of denominator. (Key point: Poles)

2. Residue at z=2:
   A. -2
   B. 2
   C. 5
   D. -5
   Answer: A. (6-4)/(2-3) = -2. (Key point: Residue at 2)

3. Residue at z=3:
   A. -5
   B. 5
   C. 2
   D. 3
   Answer: B. (9-4)/(3-2) = 5. (Key point: Residue at 3)

## Close variants set in other exams

- [Residues of a function](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aresidues-of-a-function) — Mathematics III, chapter 2, asked 3×
- [Residues of a function](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aresidues-of-a-function-two) — Mathematics III, chapter 2, asked 3×

## Related in Mathematics III

- [Residues of a function](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aresidues-of-a-function-three) — Mathematics III, chapter 2

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