Calculus

Laplace Transform

The Laplace transform sends a signal f(t) to a function F(s) of a complex variable. Plot |F(s)| as a landscape over the complex plane and the algebra turns geometric: poles are spikes, the region of convergence is a half-plane, and the Fourier transform is the slice above the imaginary axis.

the polesthe slice at σ = 0, the Fourier transformregion of convergence

Drag to orbit · scroll to zoom · height is |F(s)|, rounded off smoothly near the poles so they do not run off to infinity

Poles—

—

Signal f(t), t ≥ 0

The signal in time

Decay a 0.60

Frequency ω 2.50

Shade the region of convergenceSlice above the imaginary axis

F(s)—

region of convergence—

Fourier transform—

What to observe

Lower the decay a to zero: the pole slides onto the imaginary axis and the amber slice spikes right there. A signal that never dies out has no finite Fourier transform.
Push a negative: the signal grows, the region of convergenceretreats to the right of the axis, and the Fourier transform no longer exists. The Laplace transform still does.
On e^−atsin ωt the poles are a pair at −a ± jω, and the slice has humps at ±ω. With a small a they become a sharpresonance.
Compare e^−at with t·e^−at: same pole, but the double pole is wider near it and falls off faster away from it. A repeated pole is a factor of t in time.

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