Question

what is the difference between fourier and laplace transform

Answer 1

\(Mathematical\) Difference:

The (bilateral) Laplace Transform of \(f(t)\) is
$$
\large{
F(s) =\int_0^{\infty} e^{-st} f(t) \,dt
}
$$
where \( \large{s = \sigma + i \omega, \,} \).
The Fourier Transform of f(t) is
$$
\large{
\begin{align} \hat{f}(\omega) &= \mathcal{F}\left\{f(t)\right\} \\ &= \mathcal{L}\left\{f(t)\right\}|_{s = i\omega} = F(s)|_{s = i \omega} \\ &= \int_{-\infty}^{\infty} e^{-i \omega t} f(t)\,dt. \\\end{align}
}
$$
Comparing the two, we see that
$$
\large{
\begin{align} \hat{f}(\omega) &= \mathcal{F}\left\{f(t)\right\} \\ &= \mathcal{L}\left\{f(t)\right\}|_{s = i\omega} = F(s)|_{s = i \omega} \\ &= \int_{-\infty}^{\infty} e^{-i \omega t} f(t)\,dt. \\\end{align}
}
$$
Where \( \large{s = \sigma + i \omega, \,} \) with \(\sigma=0\) in the Fourier case.

Subjectively, Laplace uses moments as basis functions while Fourier uses frequencies. So Laplace maps a function from the time domain to it's moments and Fourier maps it to its frequencies. This is why Laplace is used in probability and Fourier is used in signal analysis.