Laplace Transform Calculator
The transform of a function of time, term by term, using the standard table and linearity.
Results
What this tool does
The Laplace transform moves a problem out of time and into a place where calculus becomes algebra: differentiating turns into multiplying, integrating turns into dividing, and a differential equation turns into something you can rearrange by hand. In practice nobody computes the defining integral — they recognise the shape and look it up. This page does exactly that, term by term, showing which table entry each piece matched, and marking plainly any term whose form is not in the table rather than inventing one.
Formula
1 → 1÷s · tⁿ → n! ÷ s^(n+1) · e^(at) → 1 ÷ (s − a) · sin(ωt) → ω ÷ (s² + ω²) · cos(ωt) → s ÷ (s² + ω²)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
expr | The expression, in t | — |
FS | The transform, in s | — |
KN | Was the form recognised? | — |
TB | Term in t · what it becomes in s | — |
NT | Terms | — |
IN | How the page read it | — |
Worked example
- The expression, in t2t^2 + 3exp(-t)
- The transform, in s4 / s^3 + 3 / (s + 1)
- Was the form recognised?Yes
- Term in t · what it becomes in s 2t^2 4 / s^3 3exp(-t) 3 / (s + 1)
- Terms2
- How the page read it2t^2 + 3exp(-t)
Limitations
- The calculation runs entirely in your browser. The values you type are never sent to a server.
- For work that must comply with a standard or be signed off, check the result against the applicable code and have it reviewed by a qualified engineer.
Frequently asked questions
Which forms does this handle?
The ones in the standard table, combined by linearity: a constant, a power of t, an exponential, a sine, a cosine, a hyperbolic sine and a hyperbolic cosine — each with a constant multiplier, and any sum or difference of them. That is how the transform is taught and how it is used in practice: nobody evaluates the defining integral, they recognise the shape and look it up. What this page will not do is the shifting and scaling theorems, convolution, or anything multiplied by t beyond the plain powers. When it meets a term it does not recognise it marks that term rather than guessing.
What is the transform for?
It turns calculus into algebra. Differentiating in time becomes multiplying by s, integrating becomes dividing by s — so a differential equation that would need integrating factors becomes an ordinary equation you can rearrange with a pencil. Solve it in the s domain, then look the answer back up in the table to get the function of time. That is why control engineering and circuit analysis live in s: a capacitor is 1/(sC), an inductor is sL, and the whole circuit becomes a resistance problem.