Fourier Transform of a Gaussian

The Fourier transform of a Gaussian is another Gaussian. This is one of the most elegant results in analysis and physics.

1. Standard Example

Let

\[ f(x) = e^{-a x^2}, \quad a > 0 \]

Using the physics convention:

\[ \mathcal{F}\{f(x)\}(k) = \int_{-\infty}^{\infty} e^{-a x^2} e^{-ikx}\, dx \]

The result is:

\[ \mathcal{F}\{e^{-a x^2}\}(k) = \sqrt{\frac{\pi}{a}} \; e^{-\frac{k^2}{4a}} \]

2. Key Insight

A Gaussian in position space transforms into a Gaussian in frequency space:

\[ e^{-a x^2} \quad \longleftrightarrow \quad e^{-\frac{k^2}{4a}} \]

• Narrow in \(x\) (large \(a\)) → Wide in \(k\)
• Wide in \(x\) (small \(a\)) → Narrow in \(k\)

This is a direct manifestation of the uncertainty principle.

3. Special Case (Unit Gaussian)

\[ f(x) = e^{-x^2} \]

\[ \mathcal{F}\{e^{-x^2}\}(k) = \sqrt{\pi}\, e^{-k^2/4} \]

4. Normalized Gaussian (Self-Fourier Form)

If we choose the symmetric normalization:

\[ f(x) = e^{-\pi x^2} \]

\[ \mathcal{F}\{f\}(k) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i k x}\, dx \]

Then:

\[ \mathcal{F}\{e^{-\pi x^2}\}(k) = e^{-\pi k^2} \]

This Gaussian is exactly its own Fourier transform.

5. Why This Is Special

The Gaussian is the only function (up to scaling and modulation) whose Fourier transform has the same functional form.

• Quantum mechanics (minimum uncertainty wave packets)
• Signal processing (Gaussian filters)
• Probability theory (normal distribution)
• Heat equation solutions