---
title: "Verify convolution theorem, f = g = e^{-x²}"
subject: "Applied Mathematics"
chapter: 4
key: "semester_4/applied_mathematics:convolution-fourier-verify-e-x2"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/convolution-fourier-verify-e-x2
site: BCT Engineering
---

# Verify convolution theorem, f = g = e^{-x²}

Applied Mathematics · Chapter 4: Fourier Transform

- Asked in: 2074 Bhadra, 5 marks
- Video: [convolution theorem || convolution theorem fourier transform || convolution theorem proof](https://www.youtube.com/watch?v=2u4JCrbqRVg) — bindas physics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/convolution-fourier-verify-e-x2
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-4

## Question

Verify the convolution theorem for Fourier transform for the functions f(x) = g(x) = e^{-x²}.

## Short answer

For f=g=e^{-x²}, F(s)=G(s)=√π e^{-s²/4} so product is π e^{-s²/2}. Their convolution is √(π/2) e^{-x²/2}, whose Fourier transform is π e^{-s²/2}. Both sides match, verifying the theorem.

## Answer

**Convention.** F{f}(s) = ∫_{−∞}^{∞} f(x) e^{−isx} dx and (f * g)(x) = ∫ f(u) g(x − u) du. The theorem states F{f * g} = F(s) · G(s). We verify it for f = g = e^{−x²}.

#### Right side: F(s) · G(s)

Using the standard result ∫ e^{−x²} e^{−isx} dx = √π e^{−s²/4}:

F(s) = G(s) = √π e^{−s²/4}

**F(s) G(s) = π e^{−s²/2}**

#### Left side: F{f * g}

**Step 1: convolution.**

```text
(f * g)(x) = ∫ e^{−u²} e^{−(x−u)²} du
```

Combine the exponents: u² + (x − u)² = 2u² − 2xu + x² = 2(u − x/2)² + x²/2. So

```text
(f * g)(x) = e^{−x²/2} ∫ e^{−2(u − x/2)²} du
           = e^{−x²/2} · √(π/2)
```

(using ∫ e^{−2t²} dt = √(π/2)).

**Step 2: transform of the convolution.** The transform of e^{−x²/2} is √(2π) e^{−s²/2}. So

```text
F{f * g} = √(π/2) · √(2π) e^{−s²/2} = π e^{−s²/2}
```

#### Comparison

Left side = π e^{−s²/2} = right side. Hence the convolution theorem is verified for f = g = e^{−x²}. ∎

(Numerical check at x = 0.8: ∫ e^{−u²−(0.8−u)²} du = 0.91009 = √(π/2) e^{−0.32} ✓.)

## Check yourself

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

1. What is F{e^{-x^2}}?
   A. sqrt(pi) e^{-s^2}
   B. pi e^{-s^2/4}
   C. sqrt(pi) e^{-s^2/4}
   D. sqrt(pi) e^{s^2/4}
   Answer: C. Standard Gaussian result. (Key point: Transform)

2. What is F(s) G(s)?
   A. sqrt(pi) e^{-s^2/4}
   B. pi e^{-s^2/4}
   C. 2 pi e^{-s^2}
   D. pi e^{-s^2/2}
   Answer: D. Product of two sqrt(pi) e^{-s^2/4}. (Key point: RHS)

3. How is the convolution evaluated?
   A. u^2 + (x-u)^2 = 2(u - x/2)^2 + x^2/2
   B. u^2 + (x-u)^2 = 2u^2
   C. u^2 + (x-u)^2 = x^2
   D. u^2 + (x-u)^2 = (u - x)^2 + x^2
   Answer: A. This leaves a Gaussian integral. (Key point: Completing square)

4. What is (f*g)(x)?
   A. sqrt(pi) e^{-x^2/2}
   B. sqrt(pi/2) e^{-x^2/2}
   C. sqrt(pi/2) e^{-x^2}
   D. pi e^{-x^2/2}
   Answer: B. The integral of e^{-2t^2} is sqrt(pi/2). (Key point: Result)

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