Fourier vs Laplace Transform Pairs

What you are seeing: the same function $f(t)$ shown alongside its Fourier-magnitude-squared $|F(\omega)|^2$ and its Laplace transform $F(s)$ on the real axis. Notice the Lorentzian shape of $|F|^2$ for an exponential decay, and the pole location in $s$ .

Figure 1. Time-domain function, Fourier power spectrum, Laplace transform.

parameter a / γ1.00

ω₀2.0

function

WHAT TO TRY

Switch the function f(t): a decaying exponential gives a Lorentzian power spectrum, a damped cosine gives a peak offset to the oscillation frequency, and each maps to a different pole pattern in the s-plane.
Tune the decay rate a or gamma: the pole moves left in the s-plane and the Fourier peak broadens, since faster decay means a wider spectral line. The pole position sets the linewidth.
Read the imaginary-axis cut: the Fourier transform is exactly the Laplace transform evaluated on s = i omega, so the |F(omega)| curve is the slice of the colour-mapped s-plane along the vertical axis.