x(t) = exp(st)

s = + j

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Mathematical Notes

Let x(t) = exp(st) where s = σ + jΩ.

    |x(t)| = exp(σ t)
    x(t) = Ω t

x(t) forms a spiral in the s-plane.

    σ < 0 : spiral in
    σ > 0 : spiral out
    σ = 0 : no spiral (circle)

    Ω < 0 : spiral clockwise
    Ω > 0 : spiral counter-clockwise
    Ω = 0 : no spiral (trace real axis)

With that, it's obvious why

    cos(Ωt) = (1/2) exp(jΩt) + (1/2) exp(-jΩt)

It's just the sum of two opposite circular
"spirals" whose imaginary parts cancel each
other out, tracing out the real axis from 
+1 to -1.

That's also why the Laplace or Fourier
transform of cos(Ωt) is two "spikes" in
the frequency domain -- because it's just
the sum of two complex exponentials.