---
title: "Residue theorem on |z|=4 with (z+1)(z+3)"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:residue-e-quantity-z1-z3"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/residue-e-quantity-z1-z3
site: BCT Engineering
---

# Residue theorem on |z|=4 with (z+1)(z+3)

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2071 Bhadra
- 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-e-quantity-z1-z3
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Evaluate the integral ∮_C [?]/((z + 1)(z + 3)) dz where C: |z| = 4, using residue theorem. [numerator unreadable in scan]

## Short answer

Assuming numerator e^z, f(z)=e^z/(( z+1)(z+3)) has simple poles at z=−1 and z=−3, both inside |z|=4. Residues are e^{−1}/2 and −e^{−3}/2. Sum = (e^{−1}−e^{−3})/2. Integral = πi(e^{−1}−e^{−3}).

## Answer

**Reading of the question:** the numerator is unreadable; the usual form of this problem is ∮_C eᶻ / ((z + 1)(z + 3)) dz with C: |z| = 4. I solve it with numerator eᶻ.

f(z) = eᶻ/((z + 1)(z + 3)). Poles are z = −1 and z = −3, both simple. Since |−1| = 1 < 4 and |−3| = 3 < 4, **both lie inside C**.

**Residue at z = −1:**

Res = eᶻ/(z + 3) at z = −1 = e⁻¹/2

**Residue at z = −3:**

Res = eᶻ/(z + 1) at z = −3 = e⁻³/(−2) = −e⁻³/2

Sum of residues = (e⁻¹ − e⁻³)/2.

By Cauchy's residue theorem:

∮_C f dz = 2πi · (e⁻¹ − e⁻³)/2

**Answer: πi (e⁻¹ − e⁻³) ≈ 0.9993 i**

(Numerically: π(0.36788 − 0.04979) = 0.9993.)

## Check yourself

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

1. Since the numerator is unreadable, what is used?
   A. z
   B. 1
   C. e^z, the usual form of the problem
   D. sin z
   Answer: C. The standard problem is ∮ e^z/((z+1)(z+3)) dz. (Key point: Assumption)

2. For |z| = 4, which poles are inside?
   A. Only z = -1
   B. Only z = -3
   C. Neither
   D. Both z=-1 and z=-3
   Answer: D. |-1| = 1 and |-3| = 3, both less than 4. (Key point: Poles inside)

3. What are the residues at -1 and -3?
   A. e^-1/2 and -e^-3/2
   B. e^-1/2 and e^-3/2
   C. e^-1 and e^-3
   D. -e^-1/2 and e^-3/2
   Answer: A. At -1: e^z/(z+3) = e^-1/2; at -3: e^z/(z+1) = -e^-3/2. (Key point: Residues)

4. What is the integral?
   A. πi(e^-1 + e^-3)
   B. πi(e^-1 - e^-3)
   C. 2πi(e^-1 - e^-3)
   D. 0
   Answer: B. 2πi × (e^-1 - e^-3)/2. (Key point: Result)

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