---
title: "Cosine transform of e^{-x}; Parseval identity"
subject: "Applied Mathematics"
chapter: 4
key: "semester_4/applied_mathematics:fourier-cosine-parseval-e-x"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/fourier-cosine-parseval-e-x
site: BCT Engineering
---

# Cosine transform of e^{-x}; Parseval identity

Applied Mathematics · Chapter 4: Fourier Transform

- Asked in: 2075 Baisakh, 5 marks; 2072 Asoj, 5 marks; 2069 Poush (asked 3 times)
- Video: [Parseval's identity on cosine Transform-Lecture 7](https://www.youtube.com/watch?v=83fn5DB0B30) — Sivanesan Gce Maths
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/fourier-cosine-parseval-e-x
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-4

## Question

Find the Fourier cosine transform of f(x) = e^{-x}, x > 0 and hence, by Parseval's identity, show that ∫₀^∞ 1/(1 + x²)² dx = π/4.

## Short answer

The cosine transform is F_c(s) = √(2/π) · 1/(1+s²). Parseval's identity gives ∫₀^∞ e^{-2x} dx = ∫₀^∞ (2/π)/(1+s²)² ds, so 1/2 = (2/π)∫₀^∞ ds/(1+s²)², hence ∫₀^∞ dx/(1+x²)² = π/4.

## Answer

**Definition used.** The Fourier cosine transform is

F_c(s) = √(2/π) ∫₀^∞ f(x) cos sx dx , with inverse f(x) = √(2/π) ∫₀^∞ F_c(s) cos sx ds.

#### Cosine transform of e⁻ˣ

Using ∫₀^∞ e^{−ax} cos sx dx = a/(a² + s²) with a = 1:

**F_c(s) = √(2/π) · 1/(1 + s²)**

#### Parseval's identity for cosine transforms

∫₀^∞ [f(x)]² dx = ∫₀^∞ [F_c(s)]² ds

**Left side:** ∫₀^∞ e^{−2x} dx = ½.

**Right side:** ∫₀^∞ (2/π) · 1/(1 + s²)² ds = (2/π) ∫₀^∞ ds/(1 + s²)².

Equating:

```text
(2/π) ∫₀^∞ ds/(1 + s²)² = 1/2
∫₀^∞ ds/(1 + s²)²       = π/4
```

Changing the dummy variable s to x:

**∫₀^∞ dx/(1 + x²)² = π/4** ∎

(Numerical check: the integral evaluates to 0.785398 = π/4.)

## Check yourself

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

1. What is the Fourier cosine transform of e^{-x}?
   A. sqrt(2/pi) s/(1 + s^2)
   B. 1/(1 + s^2)
   C. sqrt(2/pi) /(1 + s^2)
   D. sqrt(2/pi) /(1 - s^2)
   Answer: C. Using int e^{-ax} cos sx dx = a/(a^2+s^2) with a=1. (Key point: Cosine transform)

2. What is Parseval's identity for cosine transforms?
   A. int f dx = int Fc ds
   B. int f^2 dx = 2 int Fc ds
   C. int f^2 dx = int Fc ds over -infinity to infinity only
   D. int f^2 dx = int Fc^2 ds over 0 to infinity
   Answer: D. The energy is preserved under the (symmetric) cosine transform. (Key point: Parseval identity)

3. What is the value of int from 0 to infinity of e^{-2x} dx?
   A. 1/2
   B. 1
   C. 2
   D. 1/4
   Answer: A. Integral of e^{-2x} is 1/2. (Key point: LHS value)

4. How does the result pi/4 follow?
   A. (2/pi) times the integral = 1, so integral = pi/2
   B. (2/pi) times the integral = 1/2, so the integral = pi/4
   C. integral = 1/2 directly
   D. (1/pi) times the integral = 1/2, so integral = pi/2
   Answer: B. Equating gives the integral of 1/(1+s^2)^2 as pi/4. (Key point: Deduction)

## Related in Applied Mathematics

- [Cosine transform of e^{-mx}; evaluate integral](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-cosine-transform-e-mx-beta) — Applied Mathematics, chapter 4, asked 3×
- [Fourier cosine integral of e^{-kx}](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-cosine-integral-e-kx) — Applied Mathematics, chapter 4, 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×
- [Define z-transform; differ from Fourier transform](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Az-transform-vs-fourier) — Applied Mathematics, chapter 2
- [Fourier transform vs Fourier series](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-transform-vs-series) — Applied Mathematics, chapter 4
- [Sine transform of e^{-x}; ∫x²/(1+x²)²](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-sine-transform-x2-over-1-x2-sq) — Applied Mathematics, chapter 4, asked 4×

## Related in other subjects

- [Fourier cosine transform](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-cosine-transform) — 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 integral](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-cosine-integral) — Mathematics III, chapter 4
- [Fourier cosine integral](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-cosine-integral-two) — Mathematics III, chapter 4