---
title: "Contour ∫dθ/(√2-cosθ)"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:contour-1-over-sqrt2-cos"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/contour-1-over-sqrt2-cos
site: BCT Engineering
---

# Contour ∫dθ/(√2-cosθ)

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2068 Magh, 5 marks
- Video: [By Using Contour Integration Type 01 Problems 02 -Complex Integration](https://www.youtube.com/watch?v=dvwFlCOqrmA) — SK Maths Tutorial
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/contour-1-over-sqrt2-cos
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Evaluate ∫₀^{2π} dθ/(√2 - cosθ) by contour integration in complex plane.

## Short answer

Let z=e^{iθ}. Transform to 2i/(z²–2√2 z+1). Pole inside |z|=1 is z₁=√2–1, residue –1/2. Integral = 2i·2πi·(–1/2)=2π.

## Answer

**Method.** Put z = e^{iθ}, so θ: 0 → 2π maps to the unit circle |z| = 1, with

dθ = dz/(iz),  cos θ = (z² + 1)/(2z)

**Transform:**

√2 − cos θ = √2 − (z² + 1)/(2z) = (2√2 z − z² − 1)/(2z)

I = ∮_C [ 2z/(2√2 z − z² − 1) ] · dz/(iz) = ∮_C (2/i) dz/(−(z² − 2√2 z + 1)) = 2i ∮_C dz/(z² − 2√2 z + 1)

(since −2/i = 2i)

**Poles:** z² − 2√2 z + 1 = 0 gives z = √2 ± 1.

- z₁ = √2 − 1 ≈ 0.414 : **inside** C
- z₂ = √2 + 1 ≈ 2.414 : outside C

**Residue at z₁:**

Res = 1/(z₁ − z₂) = 1/((√2 − 1) − (√2 + 1)) = −1/2

**Result:** I = 2i · 2πi · (−1/2) = 2π.

**Answer: ∫₀^{2π} dθ/(√2 − cos θ) = 2π**

(Check: 2π/√(a² − b²) = 2π/√(2 − 1) = 2π.)

## Check yourself

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

1. What is cos θ in terms of z?
   A. (z²-1)/(2iz)
   B. z+1/z
   C. (z²+1)/z
   D. (z²+1)/(2z)
   Answer: D. cos θ = (z + 1/z)/2. (Key point: Substitution)

2. Which root of z²-2√2 z+1 = 0 is inside the unit circle?
   A. z = √2 - 1
   B. z = √2 + 1
   C. Both
   D. Neither
   Answer: A. √2 - 1 ≈ 0.414 inside; √2 + 1 ≈ 2.414 outside. (Key point: Poles)

3. What is the value?
   A. π
   B. 2π
   C. 2π/√2
   D. 4π
   Answer: B. 2π/√(a²-b²) = 2π/√(2-1). (Key point: Result)

## Close variants set in other exams

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