Laplace Transform Table
Searchable reference table of 40+ Laplace transform pairs covering basic functions, exponentials, trigonometric, damped sinusoids, and calculus properties
This free online laplace transform table provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Supports both metric (SI) and imperial units with built-in unit selection dropdowns on every input field, so you can work in whatever units your problem provides. Designed for engineering students and professionals working through coursework, design projects, or quick reference calculations.
Laplace Transform Reference Table
43 common Laplace transform pairs. Search or filter by category.
| Category | f(t) — Time Domain | F(s) — s Domain |
|---|---|---|
| Basic | δ(t) — Dirac delta | 1 |
| Basic | u(t) — unit step | 1/s |
| Basic | t·u(t) — ramp | 1/s² |
| Basic | tⁿ·u(t) | n! / sⁿ⁺¹ |
| Basic | t^(n-1)·u(t) / (n-1)! | 1/sⁿ |
| Basic | √(t/π) | 1 / (2·√s³) |
| Basic | 1/√(πt) | 1/√s |
| Exponential | e^(at)·u(t) | 1 / (s − a) |
| Exponential | t·e^(at)·u(t) | 1 / (s − a)² |
| Exponential | tⁿ·e^(at)·u(t) | n! / (s − a)ⁿ⁺¹ |
| Exponential | (1 − e^(−t/τ))·u(t) | 1 / (s(τs + 1)) |
| Exponential | e^(−at) − e^(−bt) | (b − a) / ((s+a)(s+b)) |
| Exponential | a·e^(−at)·u(t) | a / (s + a) |
| Trigonometric | sin(ωt)·u(t) | ω / (s² + ω²) |
| Trigonometric | cos(ωt)·u(t) | s / (s² + ω²) |
| Trigonometric | sin(ωt + φ) | (s·sinφ + ω·cosφ) / (s² + ω²) |
| Trigonometric | t·sin(ωt)·u(t) | 2ωs / (s² + ω²)² |
| Trigonometric | t·cos(ωt)·u(t) | (s² − ω²) / (s² + ω²)² |
| Trigonometric | sin²(ωt) | 2ω² / (s(s² + 4ω²)) |
| Trigonometric | cos²(ωt) | (s² + 2ω²) / (s(s² + 4ω²)) |
| Damped Sinusoid | e^(−at)·sin(ωt)·u(t) | ω / ((s+a)² + ω²) |
| Damped Sinusoid | e^(−at)·cos(ωt)·u(t) | (s+a) / ((s+a)² + ω²) |
| Damped Sinusoid | e^(−at)·t·sin(ωt) | 2ω(s+a) / ((s+a)² + ω²)² |
| Damped Sinusoid | e^(−at)·t·cos(ωt) | ((s+a)² − ω²) / ((s+a)² + ω²)² |
| Hyperbolic | sinh(at)·u(t) | a / (s² − a²) |
| Hyperbolic | cosh(at)·u(t) | s / (s² − a²) |
| Hyperbolic | e^(−at)·sinh(bt) | b / ((s+a)² − b²) |
| Control Theory | 2nd-order step: 1 − e^(−ζωnt)(cos(ωdt) + (ζ/√(1−ζ²))sin(ωdt)) | ωn² / (s(s² + 2ζωns + ωn²)) |
| Control Theory | Impulse response of 1st order: (K/τ)·e^(−t/τ) | K / (τs + 1) |
| Control Theory | PD: Kp + Kd·s | Kp/s + Kd |
| Control Theory | Integrator: 1/t | 1/s² |
| Calculus | f'(t) — first derivative | s·F(s) − f(0⁻) |
| Calculus | f''(t) — second derivative | s²·F(s) − s·f(0⁻) − f'(0⁻) |
| Calculus | ∫₀ᵗ f(τ)dτ — integral | F(s)/s |
| Calculus | t·f(t) — multiplication by t | −dF(s)/ds |
| Calculus | f(t)/t — division by t | ∫ₛ^∞ F(u)du |
| Shifting | f(t − a)·u(t − a) — time delay | e^(−as)·F(s) |
| Shifting | e^(at)·f(t) — frequency shift | F(s − a) |
| Shifting | f(at) — time scaling | (1/a)·F(s/a) |
| Special | δ(t − a) — delayed impulse | e^(−as) |
| Special | Convolution: (f*g)(t) | F(s)·G(s) |
| Special | Initial value: f(0⁺) | lim_{s→∞} s·F(s) |
| Special | Final value: f(∞) | lim_{s→0} s·F(s) |
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Laplace Transform Table. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Laplace Transform Table instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
Formula Reference
Laplace Transform Table Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
When to Use This Calculator
- •Use the Laplace Transform Table when solving homework or exam problems that require quick numerical verification of your hand calculations — instant feedback helps identify arithmetic errors before they propagate.
- •Use it during the early design phase to rapidly iterate on parameters and narrow down feasible configurations before committing time to detailed finite element simulations or full design packages.
- •Use it when reviewing a colleague's calculation or checking a vendor's data sheet for plausibility — a quick sanity check can prevent costly downstream errors.
- •Use it to generate reference data for a technical report or presentation without manual computation, ensuring consistent, reproducible numbers throughout the document.
- •Use it in the field when a quick estimate is needed and a full engineering software package is not available.
About This Calculator
The Laplace Transform Table is a precision engineering calculation tool designed for students, engineers, and technical professionals. Searchable reference table of 40+ Laplace transform pairs covering basic functions, exponentials, trigonometric, damped sinusoids, and calculus properties All calculations are performed using established engineering formulas from the relevant scientific literature and standards. Inputs support both metric (SI) and imperial unit systems, with unit conversion handled automatically — simply select your preferred unit from the dropdown next to each field. Results are computed instantly in the browser without sending data to a server, ensuring both speed and privacy. This calculator is intended as a supplementary tool for learning and design exploration; always verify results against authoritative references for safety-critical applications.
Related Calculators
Transfer Function Calculator
Analyze transfer functions: compute poles, zeros, DC gain, system order, system type, and stability from numerator/denominator coefficients
Bode Plot Generator
Generate interactive Bode plots (magnitude and phase) from transfer function coefficients with gain margin, phase margin, and crossover frequencies
Root Locus Plotter
Plot how closed-loop poles move as gain K varies; interactive K slider shows closed-loop poles, open-loop poles, and zeros
Step Response Calculator
Compute and plot unit step response for 1st and 2nd order systems with rise time, settling time, overshoot, and peak time metrics
PID Controller Tuner
Automatic PID tuning using Ziegler-Nichols (open-loop and closed-loop) and Cohen-Coon methods with step response comparison chart
Routh-Hurwitz Stability Calculator
Build the full Routh array for any-order polynomial, determine stability, and count right-half-plane poles from the characteristic equation