LaplaPy: Advanced Symbolic Laplace Transform Analysis

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Overview

LaplaPy is a professional-grade Python library for:

• Symbolic Laplace transforms with rigorous Region of Convergence (ROC) analysis
• Linear ODE solving via Laplace methods, including initial conditions
• Control system analysis: pole-zero maps, stability checks, frequency/time responses
• Educational, step-by-step output modes for teaching and self-study

Designed for engineers, scientists, and researchers in control theory, signal processing, electrical/mechanical systems, and applied mathematics.

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Features

1. Forward & Inverse Laplace with automatic ROC determination
2. ODE Solver: direct transform–solve–invert workflow for linear constant-coefficient equations
3. Pole-Zero Analysis: identify all poles and zeros symbolically
4. Stability Assessment: evaluate pole locations in the complex plane
5. Frequency Response & Bode: generate magnitude/phase plots data for ω ∈ [ω_min, ω_max]
6. Time-Domain Response: compute response to arbitrary inputs (e.g., steps, impulses, sinusoids)
7. Causal/Non-Causal Modes: model physical vs. mathematical systems
8. Step-by-Step Tracing: verbose mode to print every algebraic step (educational)

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Installation

From PyPI:

pip install LaplaPy

From source (dev):

git clone https://github.com/4211421036/LaplaPy.git
cd LaplaPy
pip install -e .[dev]

Dependencies: sympy>=1.10, numpy.

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Quickstart Examples

1. Basic Laplace Transform

from LaplaPy import LaplaceOperator, t, s

# f(t) = e^{-3t} + sin(2t)
op = LaplaceOperator("exp(-3*t) + sin(2*t)", show_steps=False)
F_s = op.forward_laplace()
print("F(s) =", F_s)       # 1/(s+3) + 2/(s^2+4)
print("ROC:", op.roc)       # Re(s) > -3

2. Inverse Transform

# Recover f(t)
f_recov = op.inverse_laplace()
print("f(t) =", f_recov)    # exp(-3*t) + sin(2*t)

3. Solve ODE

from sympy import Eq, Function, Derivative, exp

f = Function('f')(t)
ode = Eq(Derivative(f, t, 2) + 3*Derivative(f, t) + 2*f,
         exp(-t))
# ICs: f(0)=0, f'(0)=1
sol = LaplaceOperator(0).solve_ode(node,
      {f.subs(t,0):0, Derivative(f,t).subs(t,0):1})
print(sol)  # (e^{-t} - e^{-2t})

4. System Analysis & Bode Data

# H(s) = (s+1)/(s^2+0.2*s+1)
op = LaplaceOperator("(s+1)/(s**2+0.2*s+1)")
op.forward_laplace()
analysis = op.system_analysis()
# poles, zeros, stability
defp print(analysis)
# Bode data
ω, mag_db, phase = op.bode_plot(w_min=0.1, w_max=100, points=200)

5. Time-Domain Response

# Response to sin(4t)
r = op.time_domain_response("sin(4*t)")
print(r)

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CLI Usage

The LaplaPy console script provides a quick interface:

# Laplace transform + 2nd derivative:
LaplaPy "exp(-2*t)*sin(3*t)" --laplace

# Inverse Laplace:
LaplaPy "1/(s**2+4)" --inverse

# Solve ODE with ICs:
LaplaPy "f''(t)+4*f(t)=exp(-t)" --ode \
        --ic "f(0)=0" "f'(0)=1"

Flags:

• --laplace (-L) : forward transform
• --inverse (-I): inverse transform
• --ode (-O) : solve ODE
• --ic: initial conditions
• --quiet: suppress verbose steps
• --causal/--noncausal: choose system causality

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📚 Mathematical Foundations

Laplace Transform Definition

$$ \mathcal{L}{f(t)}(s) = \int_{0^-}^{\infty} e^{-st}f(t),dt $$

Derivative Property

$$ \mathcal{L}{f^{(n)}(t)}(s) = s^nF(s)-\sum_{k=0}^{n-1}s^{n-1-k}f^{(k)}(0^+) $$

Region of Convergence (ROC)

• For causal signals: $\mathrm{Re}(s) > \max(\mathrm{Re}(\text{poles}))$
• ROC ensures transform integrals converge and stability criteria

Pole-Zero & Stability

• Poles: roots of denominator $D(s)=0$
• Zeros: roots of numerator $N(s)=0$
• Stability: all poles in left-half complex plane

Frequency Response

$$ H(j\omega)=H(s)\big|_{s=j\omega} =|H(j\omega)|e^{j\angle H(j\omega)} $$

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Development & Testing

# Install dev extras
pip install -e .[dev]

# Run test suite
pytest -q

# Lint with ruff
ruff .

# Type-check (mypy)
mypy LaplaPy

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License

Distributed under the MIT License. See LICENSE for details.

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Citation

@software{LaplaPy,
  author    = {GALIH RIDHO UTOMO},
  title     = {LaplaPy: Advanced Symbolic Laplace Transform Analysis},
  year      = {2025},
  url       = {https://github.com/4211421036/LaplaPy},
  version   = {0.2.2}
}