---
title: "Fourier transform of a function"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:fourier-transform-of-a-function"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/fourier-transform-of-a-function
site: BCT Engineering
---

# Fourier transform of a function

Mathematics III · Chapter 4: Fourier Series

- 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/fourier-transform-of-a-function
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Find the Fourier transform of f(x) = 1 if |x| < 1; 0 if |x| > 1.

## Short answer

The Fourier transform is F(ω) = ∫_{-1}^1 e^(iωx) dx = 2 sin ω / ω for ω≠0, and F(0)=2. In the symmetric convention with 1/√(2π), it is √(2/π) sin ω/ω.

## Answer

The Fourier transform of f(x) is F(ω) = ∫₋∞^∞ f(x) e^(iωx) dx. (Some books use e^(−iωx) or a factor 1/√(2π); the real result below is the same up to that factor.)

For f(x) = 1 for |x| < 1 and 0 for |x| > 1:

```text
F(ω) = ∫₋₁¹ e^(iωx) dx = [ e^(iωx)/(iω) ]₋₁¹
     = (e^(iω) − e^(−iω))/(iω)
     = 2 sin ω / ω
```

**Answer:** F(ω) = 2 sin ω / ω  (ω ≠ 0), and F(0) = 2.

In the symmetric convention with the factor 1/√(2π), F(ω) = √(2/π) · sin ω / ω.

## Check yourself

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

1. F(ω) =
   A. ∫ f dx
   B. ∫0^∞ f cos ωx dx
   C. ∫ f e^(−x) dx
   D. ∫ f(x) e^(iωx) dx
   Answer: D. Given. (Key point: Definition)

2. (e^(iω) − e^(−iω))/(iω) =
   A. 2 sin ω/ω
   B. 2 cos ω/ω
   C. sin ω/ω
   D. 2 sinh ω/ω
   Answer: A. e^(iω) − e^(−iω) = 2i sin ω. (Key point: Evaluation)

3. F(0) =
   A. 0
   B. 2
   C. 1
   D. ∞
   Answer: B. Length of interval. (Key point: F(0))

## Related in Mathematics III

- [Fourier transform of a function](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-transform-of-a-function-two) — Mathematics III, chapter 4
- [Fourier sine transform](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-sine-transform) — Mathematics III, chapter 4
- [Fourier cosine transform](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-cosine-transform) — Mathematics III, chapter 4
- [Fourier series of a function](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-of-a-function) — Mathematics III, chapter 4, asked 6×
- [Inverse Z-transform of function](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-z-transform-of-function) — Mathematics III, chapter 3
- [Fourier series: x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-x2) — Mathematics III, chapter 4, asked 9×

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