Inverse Laplace Transform Calculator
Translate Laplace-domain terms into time-domain functions with core transform pairs.
Table of Contents
How to Use
- Choose the transform form you want to invert
- Enter the coefficient and required parameters (shift, ω, or power)
- Optionally set a time value to preview f(t) numerically
- Calculate to see the time-domain function and sample values
Common inverse Laplace pairs
The calculator applies standard inverse pairs directly: A/(s - a) → A·e^{at}, A/(s^2 + ω^2) → (A/ω)·sin(ωt), A·s/(s^2 + ω^2) → A·cos(ωt), and A/s^n → (A·t^{n-1})/(n-1)!. These cover many control, circuits, and mechanical responses.
- Poles at s = a drive exponential growth or decay
- Imaginary poles at ±jω produce sustained oscillations
- Repeated poles at the origin yield polynomial time terms
When to use each form
- Use A/(s - a) for first-order growth/decay responses
- Use sine or cosine forms for steady oscillations
- Use A/s^n to represent ramp, parabola, or higher-order time terms
- Combine multiple pairs with linearity when breaking down more complex transforms
Frequently Asked Questions
- Which transforms does this calculator support?
- It covers the most common single-term pairs: exponential, sine, cosine, and power-of-s terms. For more complex expressions, split them into sums of these basics and apply linearity.
- What does the time sample mean?
- The sample evaluates f(t) at the time you choose, giving a quick numeric check of the shape of the response.
- How do I handle phase shifts or delays?
- This tool focuses on core amplitude and pole/zero structure. For time delays (e^{-sT}) or phase shifts, include them separately when combining results analytically.