Chapter 42
Laplace Transform — Temporal Coherence Mapping

Flow Algebra

flow "Laplace Transform — Temporal Coherence Mapping"

# ---------- CONCEPT ----------
# The Laplace Transform generalises the Fourier Transform by extending frequency (iω) into the complex plane (s = σ + iω).
# It maps signals from the time domain into the s-domain — a space where growth, decay, and oscillation coexist as one continuous structure.
# The transform captures both amplitude and stability, encoding how coherence evolves over time through exponential weighting.

# ---------- STATES ----------
state LaplaceSystem(L):
  TimeDomainSignal:
    description: "f(t), t ≥ 0 — time-domain function representing temporal coherence or system response."
  ComplexFrequencyDomain:
    description: "F(s) — Laplace transform, describing behaviour in the complex frequency plane (s = σ + iω)."
  ForwardTransform:
    description: "F(s) = ∫₀^∞ f(t) e^{−st} dt — projection of time-domain behaviour onto exponential basis e^{−st}."
  InverseTransform:
    description: "f(t) = (1/2πi) ∫_{γ−i∞}^{γ+i∞} F(s) e^{st} ds — reconstruction of time-domain signal."
  RegionOfConvergence:
    description: "Set of s for which ∫₀^∞ |f(t)e^{−σt}|dt < ∞ — defines system stability domain."
  RelationshipToFourier:
    description: "At σ = 0, Laplace reduces to Fourier: F(iω) = ∫₀^∞ f(t)e^{−iωt}dt."

# ---------- PARAMETERS ----------
params:
  variable_time: "t ≥ 0"
  variable_complex: "s = σ + iω"
  convergence_line: "Re(s) = σ₀"
  kernel: "e^{−st}"
  closure: true

# ---------- INVARIANTS ----------
invariants:
  - "F(s) = ∫₀^∞ f(t) e^{−st} dt."
  - "f(t) = (1/2πi) ∫_{γ−i∞}^{γ+i∞} F(s) e^{st} ds."
  - "Linearity: L{a f + b g} = aF + bG."
  - "Time shift: L{f(t−t₀)u(t−t₀)} = e^{−s t₀} F(s)."
  - "Derivative: L{f'(t)} = sF(s) − f(0)."
  - "Integral: L{∫₀^t f(τ)dτ} = F(s)/s."
  - "Convolution: L{f∗g} = F(s)G(s)."
  - "Multiplication by e^{at}: L{e^{at}f(t)} = F(s−a)."
  - "Scaling: L{f(at)} = (1/a) F(s/a)."
  - "Initial and final value theorems: f(0⁺)=lim_{s→∞} sF(s), f(∞)=lim_{s→0} sF(s)."

# ---------- TRANSFORMS ----------
transforms:
  ForwardLaplace(f):
    rule: "F(s) = ∫₀^∞ f(t) e^{−st} dt."
  InverseLaplace(F):
    rule: "f(t) = (1/2πi) ∫_{γ−i∞}^{γ+i∞} F(s)e^{st} ds."
  DerivativeRule(f):
    rule: "L{f'(t)} = sF(s) − f(0)."
  IntegralRule(f):
    rule: "L{∫₀^t f(τ)dτ} = F(s)/s."
  TimeShift(f,t₀):
    rule: "L{f(t−t₀)u(t−t₀)} = e^{−s t₀} F(s)."
  FrequencyShift(f,a):
    rule: "L{e^{at}f(t)} = F(s−a)."
  ConvolutionProperty(f,g):
    rule: "L{f∗g} = F(s)G(s)."
  TransferFunction(f_in,f_out):
    rule: "H(s) = F_out(s)/F_in(s) — system coherence in s-domain."
  EvaluateSteadyState(H):
    rule: "Frequency response: H(iω) = steady oscillatory coherence."

# ---------- MONITORS ----------
monitors:
  ConvergenceRegion:
    measure: "σ₀ = Re(s)"
    meaning: "Boundary beyond which transform converges."
  StabilityCheck:
    meaning: "All poles of F(s) (roots of denominator) in Re(s) < 0 ⇒ system stable."
  CausalityCondition:
    meaning: "Signal f(t)=0 for t<0 ensures region-of-convergence is right half-plane."
  PolesZerosMap:
    meaning: "Visualise F(s): poles indicate resonant modes; zeros indicate cancellations."
  InverseContour:
    meaning: "γ chosen to right of all poles for valid inversion integral."

# ---------- EQUATIONS ----------
equations:
  - "F(s) = \\int_0^{\\infty} f(t)e^{−st}dt"
  - "f(t) = \\frac{1}{2πi}\\int_{γ−i∞}^{γ+i∞} F(s)e^{st}ds"
  - "L\\{f'(t)\\} = sF(s) − f(0)"
  - "L\\{e^{at}f(t)\\} = F(s−a)"
  - "L\\{f(t−t₀)u(t−t₀)\\} = e^{−st₀}F(s)"
  - "L\\{∫_0^t f(τ)dτ\\} = F(s)/s"
  - "L\\{f∗g\\} = F(s)G(s)"
  - "f(0⁺)=\\lim_{s→∞}sF(s),\\quad f(∞)=\\lim_{s→0}sF(s)"
  - "\\text{If } f(t)=e^{at},\\; F(s)=1/(s−a)"
  - "\\text{At } s=iω: F(iω)=\\text{Fourier transform.}"

# ---------- NOTES ----------
notes:
  - "The Laplace Transform extends Fourier analysis into the complex domain, combining oscillation (imaginary part) and exponential growth/decay (real part)."
  - "Each term e^{−st} represents a curved exponential boundary: rotation (iω) and damping (σ) unified in one operator."
  - "The real part σ determines convergence and stability — how fast coherence decays; the imaginary part ω determines oscillatory phase."
  - "Poles of F(s) reveal a system’s natural modes of response: their position in the complex plane encodes resonance, damping, and feedback strength."
  - "Laplace analysis thus transforms temporal evolution into spatial geometry — curvature of phase versus flow of time."
  - "In Flow Algebra, the Laplace Transform is a mapping from temporal coherence to complex curvature: it reveals how flow stabilises, oscillates, or decays, expressed as geometry in the s-plane — a complete map of dynamic equilibrium."