liminfoLaplace Transform Table
Free web tool: Laplace Transform Table
| f(t) | \u2194 | F(s) |
|---|---|---|
| 1 (unit step) | \u2194 | 1/s |
| t | \u2194 | 1/s² |
| tⁿ | \u2194 | n!/sⁿ⁺¹ |
| e⁻ᵃᵗ | \u2194 | 1/(s+a) |
| t e⁻ᵃᵗ | \u2194 | 1/(s+a)² |
| tⁿ e⁻ᵃᵗ | \u2194 | n!/(s+a)ⁿ⁺¹ |
| sin(ωt) | \u2194 | ω/(s²+ω²) |
| cos(ωt) | \u2194 | s/(s²+ω²) |
| e⁻ᵃᵗ sin(ωt) | \u2194 | ω/((s+a)²+ω²) |
| e⁻ᵃᵗ cos(ωt) | \u2194 | (s+a)/((s+a)²+ω²) |
| t sin(ωt) | \u2194 | 2ωs/(s²+ω²)² |
| t cos(ωt) | \u2194 | (s²-ω²)/(s²+ω²)² |
| sinh(ωt) | \u2194 | ω/(s²-ω²) |
| cosh(ωt) | \u2194 | s/(s²-ω²) |
| δ(t) (impulse) | \u2194 | 1 |
| δ(t-a) | \u2194 | e⁻ᵃˢ |
| u(t-a) (step) | \u2194 | e⁻ᵃˢ/s |
| t½ | \u2194 | √π / (2s³ˀ²) |
| 1/√(πt) | \u2194 | 1/√s |
| erfc(a/2√t) | \u2194 | e⁻ᵃ√ˢ / s |
About Laplace Transform Table
The Laplace Transform Table is a free, searchable engineering reference tool presenting 20 standard Laplace transform pairs organized into four categories: Basic (unit step, t, tⁿ, e⁻ᵃᵗ, t·e⁻ᵃᵗ, tⁿ·e⁻ᵃᵗ), Trigonometric (sin ωt, cos ωt, damped sine and cosine, t·sin ωt, t·cos ωt), Hyperbolic (sinh ωt, cosh ωt), and Special functions (Dirac delta δ(t), time-shifted delta δ(t-a), unit step u(t-a), t^½, 1/√(πt), and erfc). The tool also includes a complete Properties table covering the 11 most important Laplace transform theorems used in system analysis.
Electrical engineers, mechanical engineers, control systems designers, signal processing students, and applied mathematicians use this table when working with ordinary differential equations, circuit analysis, mechanical vibration problems, and control system transfer functions. The two-tab interface separates the Transform Pairs lookup (with category filter and search) from the Properties table, which covers linearity, time shift, frequency shift, time scaling, time derivative (differentiation), time integral, convolution, multiplication by t, division by t, initial value theorem, and final value theorem.
All data is embedded directly in the browser client — no server requests are made. The category filter uses React state with useMemo for instant filtering without any lag. Transform pairs display the time-domain function f(t) and its Laplace-domain equivalent F(s) in monospace font, clearly showing the bilateral correspondence. Unicode superscripts and special characters are used to render mathematical expressions like e⁻ᵃˢ and ω/(s²+ω²) without requiring MathJax.
Key Features
- 20 Laplace transform pairs covering Basic, Trigonometric, Hyperbolic, and Special function categories
- 11 Laplace transform properties: linearity, time/frequency shift, scaling, derivative, integral, convolution
- Initial Value Theorem and Final Value Theorem included in the properties table
- Category filter buttons for focusing on one class of transforms at a time
- Real-time search across f(t) and F(s) notation — find transforms by expression
- Two-tab layout separating Transform Pairs from Properties for clear navigation
- Monospace font display for f(t) and F(s) with Unicode math characters (ω, δ, ∫, ↔)
- Fully client-side — no server calls, works offline once the page is loaded
Frequently Asked Questions
What is the Laplace transform and why is it used in engineering?
The Laplace transform converts a time-domain function f(t) into a complex frequency-domain function F(s) = ∫₀^∞ f(t)e^(-st) dt. This transformation converts differential equations into algebraic equations, making them easier to solve. Engineers use it for circuit analysis (replacing inductors and capacitors with impedances), control system design (working with transfer functions), and mechanical vibration analysis.
What is the Laplace transform of e^(-at)?
The Laplace transform of e^(-at) is L{e^(-at)} = 1/(s+a), valid for Re(s) > -a. This is one of the most fundamental pairs because many physical systems have exponentially decaying responses. The shifted variant L{t·e^(-at)} = 1/(s+a)² and L{tⁿ·e^(-at)} = n!/(s+a)^(n+1) describe overdamped system responses.
What are the Laplace transforms of sin(ωt) and cos(ωt)?
L{sin(ωt)} = ω/(s²+ω²) and L{cos(ωt)} = s/(s²+ω²). For damped oscillations: L{e^(-at)sin(ωt)} = ω/((s+a)²+ω²) and L{e^(-at)cos(ωt)} = (s+a)/((s+a)²+ω²). These arise in second-order systems with underdamped natural frequency ω.
What is the Laplace transform of the Dirac delta function?
The Laplace transform of the Dirac delta (impulse) δ(t) is L{δ(t)} = 1. For a time-shifted impulse, L{δ(t-a)} = e^(-as). The delta function is used to model instantaneous inputs (like a hammer strike) in control systems and signal processing. Its transform being 1 means it contains all frequencies equally.
What does the Time Derivative property say?
The Time Derivative property states that L{f'(t)} = sF(s) - f(0⁺). This means differentiation in the time domain corresponds to multiplication by s in the Laplace domain, minus the initial condition f(0⁺). For second derivatives: L{f''(t)} = s²F(s) - sf(0⁺) - f'(0⁺). This property is why the Laplace transform converts differential equations into algebraic ones.
What are the Initial Value and Final Value Theorems?
The Initial Value Theorem states that f(0⁺) = lim(s→∞) s·F(s), allowing you to find the initial behavior of a system without inverting the transform. The Final Value Theorem states that f(∞) = lim(s→0) s·F(s), giving the steady-state value of a stable system. The Final Value Theorem only applies when all poles of s·F(s) have negative real parts (the system is stable).
What is the Convolution property in Laplace transforms?
The Convolution property states that if f*g(t) = ∫₀ᵗ f(τ)g(t-τ)dτ (convolution in time), then L{f*g} = F(s)·G(s) (multiplication in frequency). This is extremely useful in control systems and signal processing because convolution (which represents the output of a linear system given its impulse response) becomes simple multiplication in the Laplace domain.
How is the Laplace transform used in control system analysis?
In control systems, the Laplace transform is used to derive transfer functions H(s) = Y(s)/U(s), which describe the ratio of output Y(s) to input U(s) in the frequency domain. Engineers analyze stability using pole locations, design feedback controllers (PID), and compute step or impulse responses using the pairs and properties in this table. Inverse Laplace transform (partial fraction expansion) converts F(s) back to f(t).