Laplace Transform Calculator
Laplace Transform Result:
Common Laplace Transforms:
| ℒ{ sin(t) } | = | 1/(s^2+1) |
| ℒ{ cos(t) } | = | s/(s^2+1) |
| ℒ{ e^(-at) } | = | 1/(s+a) |
| ℒ{ t^n } | = | n!/s^(n+1) |
| ℒ{ δ(t) } | = | 1 |
| ℒ{ u(t) } | = | 1/s |
Online Laplace transform calculator converts time-domain functions to frequency-domain with dual visualization. Supports standard transforms and custom expressions with interactive charts.
What is Laplace Transform?
The Laplace transform is an integral transform that converts a function of a real variable (often time) to a function of a complex variable (frequency). It’s widely used in engineering and physics to simplify differential equations.
Key Formula
The unilateral Laplace transform is defined as:\
F(s) = ℒ{f(t)} = ∫₀^∞ f(t)e^(-st) dt
Where:
f(t)= Time-domain function (t ≥ 0)F(s)= Complex frequency-domain functions= σ + iω (complex frequency)
How to Use This Calculator
- Select Standard Function: Choose from common transforms (sine, cosine, exponential, etc.)
- Custom Mode: Enable checkbox to enter your own transform pair
- Real-time Visualization: Dual charts show both time-domain and frequency-domain representations
- Reference Table: View common transform pairs for quick lookup
FAQs
Q: What variables should I use?
A: Use t for time-domain and s for frequency-domain expressions.
Q: Can I plot discontinuous functions?
A: Yes, but use proper notation (e.g., Heaviside step function u(t) for discontinuities).
Q: How accurate are the charts?
A: Charts sample 100 data points with math.js evaluation engine.
Terminology Explained
- Time Domain: Original function representation (input)
- Frequency Domain: Transformed function representation (output)
- Region of Convergence: Range where the transform exists
- Inverse Transform: Process of recovering f(t) from F(s)
- Transfer Function: Ratio of output to input in s-domain
Supported Functions Include:
- Elementary functions (sin, cos, exp, log)
- Polynomial expressions
- Dirac delta (δ) and unit step (u) functions
- Exponential decay/growth terms