Laplace Transform Calculator
Free Laplace transform calculator with steps. Transform f(t) into F(s) using the standard table — powers, exponentials, sine, cosine, hyperbolic functions, products and the Dirac delta — with the rule for each term shown.
Type a function of t and get F(s) instantly, with the table rule that produced each term. Powers, exponentials, sines and cosines, hyperbolic functions, products like t·e^(at), and the Dirac delta are all covered — and the inverse direction is one click away.
Use t as the variable. Examples: t^3, e^(-2t), sin(3t), t*cos(2t), 4t^2+3. Write δ(t) as delta(t).
How to find a Laplace transform
- 1
Write f(t) as a sum of simple terms
The transform is linear, so a sum can be transformed one term at a time and the results added.
- 2
Match each term to a table entry
Nearly everything reduces to tⁿ, e^(at), sin(bt), cos(bt), or a product of these.
- 3
Apply the shifting rules
A factor of e^(at) replaces s with s − a; a factor of t differentiates the transform with respect to s.
- 4
Add the results
Combine the transformed terms, keeping each coefficient, to get F(s).
Laplace transform table
| f(t) | F(s) | Condition |
|---|---|---|
| 1 | 1/s | s > 0 |
| t | 1/s² | s > 0 |
| tⁿ | n!/sⁿ⁺¹ | n a positive integer |
| e^(at) | 1/(s − a) | s > a |
| sin(at) | a/(s² + a²) | s > 0 |
| cos(at) | s/(s² + a²) | s > 0 |
| sinh(at) | a/(s² − a²) | s > |a| |
| cosh(at) | s/(s² − a²) | s > |a| |
| tⁿe^(at) | n!/(s − a)ⁿ⁺¹ | s > a |
| e^(at)sin(bt) | b/((s − a)² + b²) | s > a |
| e^(at)cos(bt) | (s − a)/((s − a)² + b²) | s > a |
| t·sin(at) | 2as/(s² + a²)² | s > 0 |
| t·cos(at) | (s² − a²)/(s² + a²)² | s > 0 |
| δ(t) | 1 | all s |
| u(t) | 1/s | s > 0 |
Transforms of specific functions
Basic
Exponential
Trigonometric
Hyperbolic
Combinations
Properties you will use constantly
| Linearity | L{af + bg} = aF(s) + bG(s) |
| First derivative | L{f′} = sF(s) − f(0) |
| Second derivative | L{f″} = s²F(s) − sf(0) − f′(0) |
| First shifting (s-shift) | L{e^(at)f(t)} = F(s − a) |
| Second shifting (t-shift) | L{u(t − c)f(t − c)} = e^(−cs)F(s) |
| Multiplication by t | L{t·f(t)} = −F′(s) |
| Integration | L{∫₀ᵗ f(τ)dτ} = F(s)/s |
| Convolution | L{(f * g)(t)} = F(s)G(s) |
| Initial value theorem | f(0⁺) = lim(s→∞) sF(s) |
| Final value theorem | lim(t→∞) f(t) = lim(s→0) sF(s) |
Frequently asked questions
What is the Laplace transform? ▾
The Laplace transform converts a function of time f(t) into a function of a complex variable s, defined by F(s) = ∫₀^∞ e^(−st) f(t) dt. It turns differentiation into multiplication by s, which is what makes it so useful for differential equations.
Why use the Laplace transform to solve differential equations? ▾
Transforming a differential equation turns derivatives into algebra: dy/dt becomes sY(s) − y(0), and d²y/dt² becomes s²Y(s) − sy(0) − y′(0). You solve the resulting algebraic equation for Y(s) and then invert it, and the initial conditions are built in from the start.
What is the Laplace transform of sin(at)? ▾
L{sin(at)} = a/(s² + a²). Note the numerator is the frequency a, not s — that distinguishes it from cosine, whose transform is s/(s² + a²).
What is the region of convergence? ▾
The integral only converges for Re(s) greater than some value. For e^(at) that value is a; for a polynomial it is 0. In practice, as long as you are transforming a function that does not grow faster than an exponential, the transform exists.
What is the first shifting theorem? ▾
Multiplying f(t) by e^(at) shifts the transform: L{e^(at)f(t)} = F(s − a). It is how e^(at)sin(bt) and e^(at)cos(bt) get their transforms from the plain sine and cosine pairs.
How do I enter functions into this calculator? ▾
Use t as the variable. Write powers with ^ (t^3), exponentials as e^(2t) or exp(2t), and products with * (t*sin(2t)). Write the Dirac delta as delta(t) and the unit step as u(t).
About this Calculator
Free Laplace transform calculator with steps. Transform f(t) into F(s) using the standard table — powers, exponentials, sine, cosine, hyperbolic functions, products and the Dirac delta — with the rule for each term shown.