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

# Evaluate a line integral

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

- Asked in: 2081 Chaitra (new course), 4 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-five
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Evaluate ∮_C (sin z)/z² dz, where C is any closed path not passing through 0.

## Short answer

Expand sin z / z² = 1/z − z/6 + …; simple pole at 0 with residue 1. ∮_C = 2πi if C encloses origin, 0 if not.

## Answer

Let f(z) = sin z / z². Its only singularity is z = 0. Expand sin z:

```text
sin z / z² = (z − z³/6 + z⁵/120 − ...)/z² = 1/z − z/6 + z³/120 − ...
```

It is a simple pole with residue = coefficient of 1/z = 1.

By Cauchy's residue theorem (C simple closed, anticlockwise):

- If z = 0 lies **inside** C: ∮_C (sin z)/z² dz = 2πi × 1 = **2πi**.
- If z = 0 lies **outside** C: f is analytic inside and on C, so by Cauchy's theorem the integral is **0**.

(Alternatively, by Cauchy's integral formula for the derivative with g(z) = sin z: ∮ g(z)/z² dz = 2πi g'(0) = 2πi cos 0 = 2πi.)

**Answer: 2πi if C encloses the origin, 0 if it does not** (for a closed path winding n times, 2πi·n).

## Check yourself

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

1. sin z / z^2 has:
   A. double pole at 0
   B. removable singularity
   C. a simple pole at z=0
   D. essential singularity
   Answer: C. Expansion starts at 1/z. (Key point: Singularity)

2. The residue is:
   A. 0
   B. 1/6
   C. 2
   D. 1
   Answer: D. Coefficient of 1/z. (Key point: Residue)

3. If C encloses origin the integral is:
   A. 2 pi i
   B. 0
   C. pi i
   D. 2 pi
   Answer: A. 2 pi i times residue 1. (Key point: Inside)

4. If C does not enclose 0:
   A. 2 pi i
   B. 0, by Cauchy's theorem
   C. pi i
   D. undefined
   Answer: B. Function analytic inside and on C. (Key point: Outside)

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