The Exponential Function Learns To Wear Poles
This scene uses the classical Padé rational approximation of the exponential function. Instead of drawing arbitrary roots and poles, we start with
Rₙ(ζ) = Pₙ(ζ) / Pₙ(−ζ),
where Pₙ is the diagonal Padé numerator for eᶻ. The roots of Pₙ become the zeros of the rational function, and the roots of Pₙ(−ζ) become its poles.
Then the complex plane is gently warped by
ζ = λ(t)eⁱᶿ(t)(z c₂(t)z² c₃(t)z³ s(t))
The visible pellets are the actual preimages of the Padé zeros and poles under that map. The background comes from log|Rₙ(ζ)|, the ribbons come from arg(∂z log Rₙ), the moving mist rides
dz/dt = −1/(∂z log Rₙ),
and the faint aurora shows where the rational approximation is closest to eᶻ.