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Laplace transform studies

An exploratory notebook on making the Laplace transform feel natural. Reweight and wind a signal, move through complex frequency and reconstruct along a contour; echoes, convergence regions and time direction reveal what the extra real coordinate contributes beyond Fourier.

Frequency and decay make a landscape
OscillateDecayIntegrate
Brochure →Open project ↗
  1. σ

    Reweight time

    Exponential weighting changes which part of the signal dominates.

  2. iω

    Wind and accumulate

    Complex rotation exposes cancellation at each chosen frequency.

  3. s

    Explore the plane

    Connect poles and convergence regions to the original time behavior.

  4. ∮

    Recover the signal

    Inspect contour reconstruction and the assumptions that make it meaningful.

A work in progress about new ways to see the transform, with its mathematical assumptions kept visible.

Earlier desktop study: rotation and exponential decay form a luminous spiral.
Earlier desktop study: rotation and exponential decay form a luminous spiral.
Earlier desktop study: follow the signal through its complex integrand, Riemann sum and transform.
Earlier desktop study: follow the signal through its complex integrand, Riemann sum and transform.
Finite-band contour inversion in the current browser observatory. Original and reconstructed signals are compared with the cancelling frequency contributions; the example uses inverse bandwidth 16.
Finite-band contour inversion in the current browser observatory. Original and reconstructed signals are compared with the cancelling frequency contributions; the example uses inverse bandwidth 16.
Fourier Scanline
Fourier Scanline