---
title: "Fourier integral of rectangular pulse"
subject: "Applied Mathematics"
chapter: 4
key: "semester_4/applied_mathematics:fourier-integral-rectangular-pulse"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/fourier-integral-rectangular-pulse
site: BCT Engineering
---

# Fourier integral of rectangular pulse

Applied Mathematics · Chapter 4: Fourier Transform

- Asked in: 2074 Bhadra, 5 marks
- Video: [Fourier Transform of Basic Signals (Rectangular Function)](https://www.youtube.com/watch?v=M10lkIuJmTs) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/fourier-integral-rectangular-pulse
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-4

## Question

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

## Short answer

f(x)=1 for |x|<1, 0 for |x|>1. It is even so B(ω)=0, A(ω)=2 sin ω/ω. The Fourier integral representation is f(x) = (2/π) ∫₀^∞ (sin ω cos ωx)/ω dω. At jumps x=±1 it converges to 1/2.

## Answer

**Fourier integral formula.** For f(x) with ∫|f| dx finite:

f(x) = (1/π) ∫₀^∞ [ A(ω) cos ωx + B(ω) sin ωx ] dω

A(ω) = ∫_{−∞}^{∞} f(v) cos ωv dv, B(ω) = ∫_{−∞}^{∞} f(v) sin ωv dv.

**Here** f(x) = 1 for |x| < 1 and 0 for |x| > 1. It is an even function, so B(ω) = 0.

```text
A(ω) = ∫₋₁¹ cos ωv dv = 2 ∫₀¹ cos ωv dv = 2 sin ω / ω
```

**Fourier integral:**

**f(x) = (2/π) ∫₀^∞ (sin ω cos ωx / ω) dω**

**Convergence.** The integral converges to f(x) wherever f is continuous, and to the average of the one-sided limits at a jump:

- |x| < 1: the integral = 1
- |x| > 1: the integral = 0
- x = ±1: the integral = ½

**Consequence.** At x = 0: (2/π) ∫₀^∞ sin ω/ω dω = 1, which gives the well-known result ∫₀^∞ (sin ω/ω) dω = π/2.

## Check yourself

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

1. Why is B(w) = 0?
   A. f is odd
   B. f is zero for |x| > 1
   C. f is even
   D. f has a jump
   Answer: C. Integral of an even function times sin over symmetric limits is 0. (Key point: Which part vanishes)

2. What is A(w)?
   A. sin w / w
   B. 2 cos w / w
   C. 2 sin w
   D. 2 sin w / w
   Answer: D. A = int from -1 to 1 cos wv dv = 2 sin w/w. (Key point: A(w))

3. What does the integral converge to at x = 1?
   A. 1/2, the average of the one-sided limits
   B. 1
   C. 0
   D. Undefined
   Answer: A. At a jump the Fourier integral converges to the mean. (Key point: Value at jump)

4. What result follows at x = 0?
   A. pi
   B. int from 0 to inf sin w/w dw = pi/2
   C. 1
   D. pi/4
   Answer: B. (2/pi) times the integral equals 1. (Key point: Consequence)

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