---
title: "Residue theorem: ∮z²/(3+4z+z²) on |z|=2"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:residue-z2-over-3-4z-z2"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/residue-z2-over-3-4z-z2
site: BCT Engineering
---

# Residue theorem: ∮z²/(3+4z+z²) on |z|=2

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2070 Magh, 5 marks
- Video: [Cauchy's Residue Theorem Problem 2 / Complex Integration / Complex Variables / GATE / TANCET](https://www.youtube.com/watch?v=LVh8fRSoPbQ) — Easy Engineering Maths
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/residue-z2-over-3-4z-z2
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

State Cauchy's Residue theorem and use it to evaluate ∮_C z²/(3 + 4z + z²) dz where C is |z| = 2.

## Short answer

Cauchy's residue theorem: integral equals 2πi times sum of residues of enclosed singularities. f(z)=z²/((3+4z+z²) has simple poles at z=−1 (inside |z|=2) and z=−3 (outside). Residue at −1 is 1/2, so integral = πi.

## Answer

#### Cauchy's residue theorem
If f(z) is analytic inside and on a simple closed curve C except for a finite number of isolated singularities a₁, a₂, …, aₙ inside C, then

∮_C f(z) dz = 2πi Σ Res[f, aₖ].

#### Evaluation
f(z) = z²/(z² + 4z + 3) = z²/((z + 1)(z + 3)), C: |z| = 2.

Poles: z = −1 (|−1| = 1 < 2, inside) and z = −3 (|−3| = 3 > 2, outside). Both are simple.

Residue at z = −1:

Res = lim (z → −1) (z + 1) f(z) = z²/(z + 3) at z = −1 = 1/2

Therefore:

**Answer: ∮_C z²/(3 + 4z + z²) dz = 2πi × (1/2) = πi**

## Check yourself

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

1. What does the residue theorem state?
   A. ∮ f dz = sum of residues
   B. ∮ f dz = 2π × residues inside
   C. ∮ f dz = 2πi × residues of all singularities
   D. ∮ f dz = 2πi × sum of residues at singularities inside C
   Answer: D. Only the singularities inside C count. (Key point: Residue theorem)

2. Which poles of z²/((z+1)(z+3)) are inside |z| = 2?
   A. z = -1 only
   B. z = -3 only
   C. Both
   D. Neither
   Answer: A. |-1| = 1 < 2, |-3| = 3 > 2. (Key point: Poles inside)

3. What is the residue at z = -1?
   A. -1/2
   B. 1/2
   C. 1/4
   D. 1
   Answer: B. z²/(z+3) at -1 = 1/2. (Key point: Residue)

4. What is the integral?
   A. 2πi
   B. πi/2
   C. πi
   D. 0
   Answer: C. 2πi × 1/2 = πi. (Key point: Result)

## Close variants set in other exams

- [Residue theorem: ∮sin z / z⁶ on |z|=1](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aresidue-sinz-over-z6) — Applied Mathematics, chapter 1, asked 3×

## Related in Applied Mathematics

- [Residue theorem: ∮tan z dz on |z|=2](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aresidue-tan-z) — Applied Mathematics, chapter 1
- [Residue theorem: ∮(1-cos z)/z³ dz](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aresidue-1-cosz-over-z3) — Applied Mathematics, chapter 1
- [Cauchy theorem vs formula; evaluate integral](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acif-distinguish-e-z-z-1-z-2) — Applied Mathematics, chapter 1, asked 4×
- [Residue theorem on |z|=4 with (z+1)(z+3)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aresidue-e-quantity-z1-z3) — Applied Mathematics, chapter 1
- [Pole of order m; ∮cot z dz on |z|=1](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aresidue-cot-z) — Applied Mathematics, chapter 1
- [Evaluate ∫(x²+iy)dz along y = x²](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acomplex-integral-path-y-x2) — Applied Mathematics, chapter 1

## Related in other subjects

- [Evaluate using Gauss' divergence theorem F=2xi+3yj+4zk](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-gauss-divergence-theorem-f-three) — Mathematics III, chapter 2, asked 4×
- [Evaluate using Gauss' divergence theorem F=2xi+3yj+4zk](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-gauss-divergence-theorem-f-seven) — Mathematics III, chapter 2, asked 4×
- [Evaluate using Stokes' theorem](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-stokes-theorem-four) — Mathematics III, chapter 2
- [Evaluate using Stokes' theorem](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-stokes-theorem-five) — Mathematics III, chapter 2