Yes, the Fourier transform of the Cauchy distribution does indeed have the form $e^{-|W|}$. Here's the detailed explanation:

Cauchy Distribution

The Cauchy distribution, also known as the Lorentz distribution, has the probability density function (PDF):

$f(x) = \frac{1}{\pi(1 + x^2)}$

This is the standard Cauchy distribution, centered at 0 with a scale parameter of 1.

Fourier Transform

The Fourier transform of a function $f(x)$ is given by:

$\mathcal{F}\{f(x)\}(W) = \int_{-\infty}^{\infty} f(x) e^{-i W x} dx$

Substitute the PDF of the Cauchy distribution into this equation:

$\mathcal{F}\left\{ \frac{1}{\pi(1 + x^2)} \right\}(W) = \int_{-\infty}^{\infty} \frac{1}{\pi(1 + x^2)} e^{-i W x} dx$

Solving the Integral

This is a well-known integral that can be solved through complex analysis or by recognizing that it appears in standard Fourier transform tables. The result of this integral is:

$\mathcal{F}\left\{ \frac{1}{\pi(1 + x^2)} \right\}(W) = e^{-|W|}$

Thus, the Fourier transform of the standard Cauchy distribution is $e^{-|W|}$, which shows that the distribution is highly localized in both the time and frequency domains.

Key Takeaways:

• The Cauchy distribution is a heavy-tailed distribution.
• Its Fourier transform is a simple exponential function, $e^{-|W|}$, which decays as the frequency $W$ increases.

Would you like more details on the derivation or any specific part of this topic?

Here are some related questions for further exploration:

1. How does the Fourier transform behave for other probability distributions like Gaussian?
2. What is the inverse Fourier transform of $e^{-|W|}$?
3. How does the Cauchy distribution compare to the normal distribution?
4. Can we generalize the Fourier transform of the Cauchy distribution with different parameters?
5. How does the Fourier transform apply in signal processing for real-world applications?

Tip: The Cauchy distribution does not have a mean or variance due to its heavy tails, which is a key difference from the Gaussian distribution.