LaplaPy 0.1.9

Symbolic derivative & Laplace transform with step-by-step output

LaplaPy: Advanced Symbolic Laplace Transform Analysis

Scientific Computing Package for Differential Equations, System Analysis, and Control Theory
A comprehensive Python library for symbolic Laplace transforms with rigorous mathematical foundations, designed for engineers, scientists, and researchers.

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Overview

LaplaPy provides a powerful symbolic computation environment for:

1. Time-domain analysis: Derivatives, integrals, and function manipulation
2. Laplace transforms: With rigorous Region of Convergence (ROC) determination
3. System analysis: Pole-zero identification, stability analysis, and frequency response
4. ODE solving: Complete solution of linear differential equations with initial conditions
5. Control system tools: Bode plots, time-domain responses, and transfer function analysis

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Key Features

• Mathematical Rigor: Implements Laplace transform theory with proper ROC analysis
• Causal System Modeling: Automatic handling of Heaviside functions for physical systems
• Step-by-Step Solutions: Educational mode for learning complex concepts
• Comprehensive System Analysis: Pole-zero identification, stability criteria, frequency response
• ODE Solver: Complete solution workflow for linear differential equations
• Visualization Tools: Bode plot generation and time-domain simulations

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Installation

pip install LaplaPy

For development:

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

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Quickstart

Basic Operations

from LaplaPy import LaplaceOperator, t, s

# Initialize with expression (causal system by default)
op = LaplaceOperator("exp(-3*t) + sin(2*t)", show_steps=True)

# Compute derivative
d1 = op.derivative(order=1)

# Laplace transform with ROC analysis
F_s, roc, poles, zeros = op.laplace()

# Inverse Laplace transform
f_t = op.inverse_laplace()

ODE Solving

from sympy import Eq, Function, Derivative, exp

# Define a differential equation
f = Function('f')(t)
ode = Eq(Derivative(f, t, t) + 3*Derivative(f, t) + 2*f, exp(-t))

# Solve with initial conditions
solution = op.solve_ode(ode, {0: 0, 1: 1})  # f(0)=0, f'(0)=1

System Analysis

# Frequency response
magnitude, phase = op.frequency_response()

# Time-domain response to input
response = op.time_domain_response("sin(4*t)")

# Generate Bode plot data
omega, mag_db, phase_deg = op.bode_plot(ω_range=(0.1, 100), points=100)

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CLI Usage

LaplaPy "exp(-2*t)*sin(3*t)" --laplace --deriv 2
LaplaPy "s/(s**2 + 4)" --inverse
LaplaPy "Derivative(f(t), t, t) + 4*f(t) = exp(-t)" --ode --ic "f(0)=0" "f'(0)=1"

Options:

• --deriv N: Compute Nth derivative
• --laplace: Compute Laplace transform
• --inverse: Compute inverse Laplace transform
• --ode: Solve ODE (provide equation)
• --ic: Initial conditions (e.g., "f(0)=0", "f'(0)=1")
• --causal/--noncausal: System causality assumption
• --quiet: Suppress step-by-step output

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Mathematical Foundations

Laplace Transform

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

Derivative Property

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

Region of Convergence

• For causal systems: Re(s) > σ_max (right-half plane)
• Proper ROC determination for stability analysis

Pole-Zero Analysis

• Transfer function: $H(s) = \frac{N(s)}{D(s)}$
• Poles: Roots of denominator polynomial
• Zeros: Roots of numerator polynomial

Frequency Response

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

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Examples

Second-Order System Analysis

op = LaplaceOperator("1/(s**2 + 0.6*s + 1)", show_steps=True)

# Get poles and zeros
F_s, roc, poles, zeros = op.laplace()

# Frequency response
magnitude, phase = op.frequency_response()

# Bode plot data
omega, mag_db, phase_deg = op.bode_plot(ω_range=(0.1, 10), points=200)

Circuit Analysis (RLC Network)

# Define circuit equation: L*di/dt + R*i + 1/C*∫i dt = V_in
L, R, C = 0.5, 4, 0.25
op = LaplaceOperator("V_in(s)", show_steps=True)

# Impedance representation
Z = L*s + R + 1/(C*s)
current = op.time_domain_response("V_in(s)/" + str(Z))

# Response to step input
step_response = current.subs("V_in(s)", "1/s")

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Development & Testing

# Run tests
pytest tests/

# Generate documentation
cd docs
make html

# Contribution guidelines
CONTRIBUTING.md

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Scientific Applications

1. Control Systems: Stability analysis, controller design
2. Circuit Analysis: RLC networks, filter design
3. Vibration Engineering: Damped oscillator analysis
4. Signal Processing: System response characterization
5. Communication Systems: Filter design, modulation analysis
6. Mechanical Systems: Spring-mass-damper modeling

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Documentation Wiki

Full documentation available at:
LaplaPy Documentation WiKi

Includes:

• Mathematical background
• API reference
• Tutorial notebooks
• Application examples

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License

MIT License

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Cite This Work

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