Laplace Transform Calculator
Type a function of t and this calculator finds its Laplace transform F(s) instantly.
Use t as the time variable, ^ for powers, e.g. t^2, exp(-2t) or sin(3t).
How to find a Laplace transform by hand
- Write the definition: F(s) = ∫₀^∞ f(t)e⁻ˢᵗ dt.
- For common functions (powers of t, exponentials, sin, cos), look up the standard transform pairs rather than integrating from scratch.
- Break a sum of terms into separate transforms and add the results (the transform is linear).
- For a function multiplied by e^(at), shift s to (s − a) in the transform of the rest.
Exemple résolu
Transform f(t) = t². Using the standard pair for tⁿ, the transform is n!/sⁿ⁺¹ with n = 2, giving 2!/s³ = 2/s³.
See every formula worked through on its own page in the maths formula list, or differentiate a function first in the Derivative Calculator.
Foire aux questions
What is a Laplace transform used for?
It converts a function of time, f(t), into a function of a complex variable s, F(s) — turning differential equations into algebra, which is why it is used throughout engineering and physics.
What is the Laplace transform of a constant?
The Laplace transform of the constant 1 is 1/s. More generally, the transform of a constant c is c/s.
What is the Laplace transform of t²?
2/s³ (in general, the transform of tⁿ is n!/sⁿ⁺¹).
What is the Laplace transform of sin and cos?
The transform of sin(at) is a/(s² + a²), and the transform of cos(at) is s/(s² + a²).
