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

# Fourier transform of a function

Mathematics III · Chapter 4: Fourier Series

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

## Question

Find the Fourier transform of f(x) = e^(−ax) and use the result to show ∫_0^∞ dx/((a² + x²)(b² + x²)) = π/(2ab(a + b)).

## Short answer

The Fourier transform of e^(-a|x|) is 2a/(a²+ω²). Applying Parseval’s identity to f and g(x)=e^(-b|x|) gives ∫_{-∞}^∞ 4ab/[(a²+ω²)(b²+ω²)] dω = 2π ∫ e^{-(a+b)|x|} dx = 4π/(a+b). Thus ∫_0^∞ dx/[(a²+x²)(b²+x²)] = π/[2ab(a+b)].

## Answer

**Reading of the question.** The function is taken as f(x) = e^(−a|x|) with a > 0 (so that its transform exists on the whole line and the Parseval identity applies). Use F(ω) = ∫₋∞^∞ f(x) e^(iωx) dx.

**Fourier transform**

```text
F(ω) = ∫₀^∞ e^(−ax) e^(iωx) dx + ∫₋∞^0 e^(ax) e^(iωx) dx
     = 1/(a − iω) + 1/(a + iω)
     = 2a / (a² + ω²)
```

Likewise, for g(x) = e^(−b|x|), G(ω) = 2b/(b² + ω²).

**Parseval's (Plancherel) identity.** For real even functions: ∫₋∞^∞ F(ω)G(ω) dω = 2π ∫₋∞^∞ f(x)g(x) dx.

```text
Right side : 2π ∫₋∞^∞ e^(−(a+b)|x|) dx = 2π · 2/(a + b) = 4π/(a + b)
Left side  : 4ab ∫₋∞^∞ dω /[(a² + ω²)(b² + ω²)] = 8ab ∫₀^∞ dω /[(a² + ω²)(b² + ω²)]
```

Equating: 8ab ∫₀^∞ dω /[(a² + ω²)(b² + ω²)] = 4π/(a + b).

**Answer:** ∫₀^∞ dx / [(a² + x²)(b² + x²)] = π / [2ab(a + b)]

## Check yourself

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

1. Why is f(x) = e^(-ax) taken as e^(-a|x|)?
   A. Because e^(-ax) is odd
   B. Because a is negative
   C. To make the function periodic
   D. So the transform exists on the whole line and Parseval can be used
   Answer: D. e^(-ax) would blow up for negative x; e^(-a|x|) is integrable over the whole line. (Key point: Interpreting the function)

2. What is the Fourier transform of e^(-a|x|)?
   A. 2w/(a^2 + w^2)
   B. 1/(a^2 + w^2)
   C. 2a/(a^2 + w^2)
   D. a/(a^2 + w^2)
   Answer: C. 1/(a - iw) + 1/(a + iw) = 2a/(a^2 + w^2). (Key point: Transform result)

3. Using f = e^(-a|x|), g = e^(-b|x|), what is the right side 2*pi * integral of f g dx?
   A. 2 pi/(a+b)
   B. pi/(a+b)
   C. 4 pi/(a+b)
   D. 4 pi/(a-b)
   Answer: C. The integral of e^(-(a+b)|x|) over the line is 2/(a+b), times 2 pi gives 4 pi/(a+b). (Key point: Parseval value)

4. Equating 8ab * I = 4 pi/(a+b) gives I equal to?
   A. pi/[4ab(a+b)]
   B. pi/[2ab(a+b)]
   C. 2 pi/[ab(a+b)]
   D. pi/[ab(a+b)]
   Answer: B. I = 4 pi/[8ab(a+b)] = pi/[2ab(a+b)]. (Key point: Final step)

## 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) — 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×
- [Half range sine series: e^(ax)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-e-ax) — Mathematics III, chapter 4, asked 5×
- [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

## Related in other subjects

- [DUIS: ∫ (e^(−ax) − e^(−bx))/x dx](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aduis-e-ax-minus-e-bx) — Mathematics I, chapter 2, asked 9×
- [DUIS: ∫ e^(−ax) sin x / x dx](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aduis-e-ax-sinx-over-x) — Mathematics I, chapter 2, asked 8×
- [DUIS: ∫ tan⁻¹(ax)/(x(1+x²)) dx](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aduis-atan-ax-over-x-1-plus-x2) — Mathematics I, chapter 2, asked 2×
- [Fourier transform of 1-x²; evaluate integral](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-transform-1-x2-evaluate) — Applied Mathematics, chapter 4, asked 2×