Laplace Transform Calculator
Find Laplace transforms for basic time functions.
Table of Contents
How to Use
- Choose the time function type: exponential, sine, cosine, or polynomial power of t.
- Enter the coefficient and any required parameters (rate a, frequency ω, or power n).
- Set a positive value for s to sample the transform numerically.
- Run the calculation to see F(s), the time-domain expression, and the evaluated value at s.
Common Laplace transform pairs
The Laplace transform maps time-domain functions f(t) into F(s) in the complex frequency domain. It simplifies differential equations by turning derivatives into algebraic terms.
- A·e^{at} → A / (s - a)
- A·sin(ωt) → A·ω / (s^2 + ω^2)
- A·cos(ωt) → A·s / (s^2 + ω^2)
- A·t^n → A·n! / s^{n+1}
How this calculator evaluates F(s)
- Select a function type and provide its parameters.
- Build the symbolic F(s) expression from the known transform pair.
- Sample F(s) at the chosen positive s value.
- Report the qualitative behavior (growth, decay, oscillation, or polynomial).
These closed-form transforms cover common signals and are a starting point before using tables or partial fractions for more complex inputs.
Frequently Asked Questions
- Why must s be positive?
- The Laplace transform is defined for values of s where the integral converges. Requiring s > 0 reflects the standard region of convergence for these basic functions.
- Can I model damping or growth?
- Yes. Use the exponential option with a negative rate for decay or positive for growth. The behavior label highlights the trend.
- How do I handle more complex functions?
- Use linearity and known transform tables to break complex expressions into simpler parts. This calculator focuses on the most common building blocks.