Laplace Transform Table — Inverse Pairs, Properties & Cheat Sheet
The complete reference for every standard Laplace and inverse Laplace transform pair, plus all operational properties. Print-friendly, bookmark-ready.
This page is designed to be your single, authoritative reference for every standard Laplace transform pair and inverse Laplace table entry you’ll need in differential equations, control systems, signals & systems, and engineering math. Every pair is presented in clean, printable HTML — bookmark this page or print it as a one-page cheat sheet to keep alongside your homework. The inverse laplace table below covers all ~20 standard pairs, followed by the complete Laplace transform properties table covering linearity, differentiation, time-shifting, frequency-shifting, scaling, and convolution.
If you need to solve a specific F(s) expression with full step-by-step partial fraction decomposition, use our companion Inverse Laplace Transform Calculator alongside this cheat sheet — the calculator does the algebra, and this table tells you which row each result comes from.
Laplace Transform Pairs — The Complete Table
The table below contains every standard Laplace transform pair used in engineering and applied mathematics. Each row gives you both directions: read left-to-right for the forward Laplace transform ℒ{f(t)} = F(s), or read right-to-left for the inverse Laplace transform ℒ⁻¹{F(s)} = f(t). This inverse laplace table is the most-used reference in differential equations and control theory.
Table 1: Standard Laplace and Inverse Laplace Transform Pairs
| # | f(t) — Time Domain | F(s) = ℒ{f(t)} — s-Domain | Region of Convergence |
|---|---|---|---|
| 1 | 1 (unit constant) | 1/s | s > 0 |
| 2 | t | 1/s² | s > 0 |
| 3 | t² | 2/s³ | s > 0 |
| 4 | tⁿ (n = positive integer) | n! / sⁿ⁺¹ | s > 0 |
| 5 | t^a (a > −1, real) | Γ(a+1) / s^(a+1) | s > 0 |
| 6 | e^(at) | 1 / (s − a) | s > a |
| 7 | t · e^(at) | 1 / (s − a)² | s > a |
| 8 | tⁿ · e^(at) | n! / (s − a)ⁿ⁺¹ | s > a |
| 9 | sin(at) | a / (s² + a²) | s > 0 |
| 10 | cos(at) | s / (s² + a²) | s > 0 |
| 11 | sinh(at) | a / (s² − a²) | s > |a| |
| 12 | cosh(at) | s / (s² − a²) | s > |a| |
| 13 | e^(at) · sin(bt) | b / ((s − a)² + b²) | s > a |
| 14 | e^(at) · cos(bt) | (s − a) / ((s − a)² + b²) | s > a |
| 15 | t · sin(at) | 2as / (s² + a²)² | s > 0 |
| 16 | t · cos(at) | (s² − a²) / (s² + a²)² | s > 0 |
| 17 | t · sinh(at) | 2as / (s² − a²)² | s > |a| |
| 18 | t · cosh(at) | (s² + a²) / (s² − a²)² | s > |a| |
| 19 | δ(t) (Dirac delta) | 1 | all s |
| 20 | δ(t − c) | e^(−cs) | all s |
| 21 | u(t) (unit step) | 1/s | s > 0 |
| 22 | u(t − c) (shifted step) | e^(−cs) / s | s > 0 |
| 23 | (1/√(πt)) · e^(−a²/4t) | e^(−a√s) / √s | s > 0 |
| 24 | (1 − cos(at)) / a² | 1 / (s(s² + a²)) | s > 0 |
| 25 | (at − sin(at)) / a³ | 1 / (s²(s² + a²)) | s > 0 |
Laplace Transform Properties Table
While the pairs table tells you the Laplace transform of specific functions, the Laplace properties table tells you how operations (differentiation, integration, shifting, scaling, convolution) affect the transform. These operational properties allow you to compute Laplace transforms of more complex expressions by manipulating known pairs. Together, the pairs table and the properties table cover virtually every Laplace transform problem you’ll encounter.
Table 2: Operational Properties of the Laplace Transform
| Property | Time Domain — f(t) | Laplace Domain — F(s) |
|---|---|---|
| Linearity | a·f(t) + b·g(t) | a·F(s) + b·G(s) |
| First Derivative | f ‘(t) | s·F(s) − f(0) |
| Second Derivative | f ”(t) | s²·F(s) − s·f(0) − f ‘(0) |
| nth Derivative | f ⁽ⁿ⁾(t) | sⁿ·F(s) − sⁿ⁻¹·f(0) − … − f ⁽ⁿ⁻¹⁾(0) |
| Integration | ∫₀ᵗ f(τ) dτ | F(s) / s |
| Time Shift (2nd Shift Theorem) | f(t − c) · u(t − c) | e^(−cs) · F(s) |
| Frequency Shift (1st Shift Theorem) | e^(at) · f(t) | F(s − a) |
| Time Scaling | f(a·t), a > 0 | (1/a) · F(s/a) |
| Multiplication by t | t · f(t) | −F ‘(s) |
| Multiplication by tⁿ | tⁿ · f(t) | (−1)ⁿ · F⁽ⁿ⁾(s) |
| Division by t | f(t) / t | ∫ₛ^∞ F(σ) dσ |
| Convolution Theorem | (f * g)(t) = ∫₀ᵗ f(τ)·g(t−τ) dτ | F(s) · G(s) |
| Initial Value Theorem | lim(t→0⁺) f(t) | lim(s→∞) s·F(s) |
| Final Value Theorem | lim(t→∞) f(t) | lim(s→0) s·F(s) |
| Periodic Function (period T) | f(t) with f(t+T) = f(t) | [∫₀ᵀ e^(−st)·f(t) dt] / (1 − e^(−sT)) |
How to Use This Table to Find the Inverse Laplace Transform
The inverse laplace table above gives you the answer directly when your F(s) matches one of the listed forms. In practice, most F(s) expressions don’t match a row exactly — they need to be manipulated first. Here’s the practical workflow:
- Try direct matching first. Look at your F(s) and scan the inverse laplace table for a row whose right column matches your expression. If you find a direct match, the answer is the corresponding f(t) in the left column.
- Check if a property applies. If your F(s) looks like a “shifted” or “scaled” version of a known form, use the Laplace transform properties table. For example, F(s−a) means you have a frequency shift — apply the inverse and multiply by e^(at).
- Decompose with partial fractions. If F(s) is a rational function (polynomial divided by polynomial), perform partial fraction decomposition to break it into simpler terms that match table rows. This is the most common technique for solving differential equations.
- Complete the square for quadratic denominators. If you have (s² + bs + c) in the denominator with complex roots, complete the square to get the form (s−a)² + b², then match to entries 13–14 in the pairs table.
- Apply linearity. Take the inverse Laplace transform of each decomposed term separately, then sum the results. Linearity is the most-used property in inverse Laplace work.
Common Laplace Transform Pairs Explained
These are the six most-searched, most-asked-about Laplace transform pairs. Understanding these intuitively will let you solve the vast majority of textbook problems without constantly looking up the table.
ℒ{c} = c/s
Inverse Laplace of a constant. The Laplace transform of any constant c is simply c/s. This follows from ℒ{1} = 1/s and linearity. Conversely, ℒ⁻¹{c/s} = c. This is the most fundamental pair in the table.
ℒ⁻¹{1/s²} = t
Inverse Laplace of 1/s². The inverse Laplace transform of 1/s² is t. This generalizes to ℒ⁻¹{n!/sⁿ⁺¹} = tⁿ, so ℒ⁻¹{1/s³} = t²/2, and so on. Any negative power of s corresponds to a polynomial in t.
ℒ{e^(at)} = 1/(s−a)
Exponential pair. The Laplace transform of an exponential e^(at) is 1/(s−a). This is the foundation of the frequency shift property: any time you see a denominator (s−a), you’ll have an exponential factor e^(at) in the inverse.
ℒ{sin(at)} = a/(s²+a²)
Sine pair. The Laplace transform of sin(at) is a/(s²+a²). Notice the constant a in the numerator — students often forget this and write 1/(s²+a²) by mistake. The denominator pattern s²+a² always signals a trig function in the inverse.
ℒ{cos(at)} = s/(s²+a²)
Cosine pair. The Laplace transform of cos(at) is s/(s²+a²). The s in the numerator distinguishes cosine from sine. Together, sin and cos pairs cover all undamped oscillation problems in differential equations.
ℒ{δ(t)} = 1
Dirac delta pair. The Laplace transform of the Dirac delta function δ(t) is exactly 1. This makes the delta function the multiplicative identity in the s-domain — convolving any function with δ(t) leaves it unchanged.
Frequently Asked Questions
What is the Laplace transform of a constant?
The Laplace transform of a constant c is c/s. This comes from the basic pair ℒ{1} = 1/s combined with linearity: ℒ{c} = c·ℒ{1} = c/s. The region of convergence is s > 0. This is row 1 of the Laplace transform pairs table above.
What is the inverse Laplace transform of 1/s²?
The inverse Laplace transform of 1/s² is t. This is one of the most fundamental entries in any inverse laplace table. More generally, ℒ⁻¹{n!/sⁿ⁺¹} = tⁿ, so 1/s³ inverts to t²/2, 1/s⁴ inverts to t³/6, and so on.
How do you use a Laplace transform table?
To use a Laplace transform table, first identify whether you’re doing a forward transform (f(t) → F(s)) or an inverse transform (F(s) → f(t)). Match your function to a row in the table. If your function doesn’t match directly, use the Laplace transform properties table to apply linearity, time-shifting, frequency-shifting, or scaling. For complex rational F(s) expressions, perform partial fraction decomposition first to break them into recognizable forms.
What’s the difference between the transform table and the properties table?
The transform pairs table lists specific f(t) ↔ F(s) relationships for common functions like sin(at), e^(at), and tⁿ. The Laplace properties table lists operational rules — how operations like differentiation, integration, time-shifting, and convolution affect Laplace transforms. You use the pairs table for direct lookup and the properties table to manipulate transforms algebraically when no direct match exists.
Can I print this Laplace transform cheat sheet?
Yes — this page is specifically designed as a print-friendly Laplace transform sheet. Click the “Print this Table” button above either table, or use your browser’s print function (Ctrl+P / Cmd+P). The print stylesheet automatically hides navigation elements and formats the tables for clean single-page printing, making this work just as well as a downloadable Laplace transform cheat sheet PDF.