---
title: "Evaluate a line integral"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:evaluate-a-line-integral-four"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-a-line-integral-four
site: BCT Engineering
---

# Evaluate a line integral

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

- Asked in: 2081 Chaitra (new course), 2 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-a-line-integral-four
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Show that ∮_C dz/(z − a) = 2πi where C is the unit circle |z − a| = 1.

## Short answer

Set z-a = e^(iθ). Then dz = i e^(iθ) dθ, so the integral becomes ∫_0^{2π} i dθ = 2πi, consistent with Cauchy's integral formula for f(z)=1.

## Answer

On the circle |z − a| = 1, put z − a = e^(iθ), so z = a + e^(iθ), 0 ≤ θ ≤ 2π. Then dz = i e^(iθ) dθ.

```text
∮_C dz/(z − a) = ∫₀^2π (i e^(iθ) dθ) / e^(iθ)
               = i ∫₀^2π dθ
               = 2πi
```

**Hence ∮_C dz/(z − a) = 2πi.** (This also follows from Cauchy's integral formula with f(z) = 1, as the point a lies inside C.)

## Check yourself

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

1. On |z-a|=1 put:
   A. z = e^(i theta)
   B. z - a = e^(i theta)
   C. z - a = theta
   D. z = a e^(i theta)
   Answer: B. Circle centred at a. (Key point: Parametrization)

2. dz equals:
   A. e^(i theta) d theta
   B. i d theta
   C. i e^(i theta) d theta
   D. -i e^(i theta) d theta
   Answer: C. Differentiate. (Key point: dz)

3. Integral gives:
   A. 2 pi
   B. 0
   C. pi i
   D. 2 pi i
   Answer: D. i times integral of dtheta over 0 to 2pi. (Key point: Result)

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