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

# Residues of a function

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

- Asked in: 2082 Kartik (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-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Find the residue of f(z) = (3z − 4)/((z − 1)(z − 2)) at each of its singularities.

## Short answer

f(z) has simple poles at z=1 and z=2. The residue at z=1 is 1 and at z=2 is 2, giving the partial fractions 1/(z-1) + 2/(z-2).

## Answer

f(z) = (3z − 4)/((z − 1)(z − 2)) has simple poles at z = 1 and z = 2.

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

**Answer:** residue = 1 at z = 1 and 2 at z = 2. (Check by partial fractions: f = 1/(z − 1) + 2/(z − 2).)

## Check yourself

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

1. Poles of (3z-4)/((z-1)(z-2)) are:
   A. Simple poles at 1 and 2
   B. Double pole at 1
   C. Simple pole at 0
   D. Poles at -1,-2
   Answer: A. Distinct linear factors. (Key point: Poles)

2. Residue at z=1:
   A. -1
   B. 1
   C. 2
   D. 3
   Answer: B. (3-4)/(1-2) = 1. (Key point: Residue at 1)

3. Residue at z=2:
   A. 1
   B. -2
   C. 2
   D. 4
   Answer: C. (6-4)/(2-1) = 2. (Key point: Residue at 2)

## 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-four) — 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

## Related in other subjects

- [Zero of order m; poles and residues](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Azero-order-m-poles-residue) — Applied Mathematics, chapter 1
- [Poles and residues of z²/((z-1)²(z+2))](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Apoles-residues-z2-z1sq-z2) — Applied Mathematics, chapter 1
- [Controllable canonical form of (z⁻¹+2z⁻²)/(1+4z⁻¹+3z⁻²)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Accf-z1-2z2-4z1-3z2) — Digital Control System, chapter 5, asked 15×
- [Controllable canonical form of (4z³+3z²+5z+4)/(2z³+5z²+2z+3)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Accf-4z3-3z2) — Digital Control System, chapter 5, asked 14×