Laplace Transform Calculator

Compute Laplace transforms and inverse Laplace transforms for standard f(t) and rational F(s). Table-based, with partial fractions for inverses.

Mode

Function

F(s) = L{f(t)}

3 / (((s + 2))^2 + 9)

Method1 term transformed with the standard Laplace table.

ROCTypically Re(s) greater than the rightmost pole

How to use this laplace transform calculator

Enter f(t) with t as the time variable: polynomials, exp(at) or e^(at), sin, cos, sinh, cosh, and products such as t*exp(-t)*cos(2t). Multiplication can be written with * or implied (2t, 3sin(t)). For inverse mode, enter a rational F(s) such as (s+1)/(s^2+2s+5).

This is a purpose-built Laplace transform calculator for differential-equations courses—not a full computer-algebra kernel. If a form is missing from the table, expand it into a sum of supported terms. Unit steps u(t−a) and impulses delta(t−a) are supported on the forward transform.

Formula

L{f(t)} = ∫₀^∞ e^{−st} f(t) dt. Standard pairs: L{tⁿ} = n!/s^{n+1}, L{e^{at}} = 1/(s−a), L{sin(ωt)} = ω/(s²+ω²), L{cos(ωt)} = s/(s²+ω²), L{e^{at}sin(ωt)} = ω/((s−a)²+ω²). Inverse of a proper rational F(s) uses partial fractions plus the same table.

Worked example

L{e^{−2t} sin(3t)} = 3 / ((s+2)² + 9) = 3/(s² + 4s + 13). Inverse of 1/(s²+1) is sin(t).

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Frequently asked questions

How do I take a Laplace transform?

Rewrite f(t) as a sum of table functions (polynomials, exponentials, sin/cos) and apply linearity. This tool matches those patterns automatically.

What is the inverse Laplace transform?

The unique causal f(t) whose transform is F(s). For rational F(s), decompose into partial fractions and invert term by term.

Does this handle e^{at} sin(bt)?

Yes. L{e^{at} sin(bt)} = b / ((s−a)² + b²), and the cosine pair uses (s−a) in the numerator.

Can I type 1/(s^2+1) for inverse?

Yes. Choose Inverse Laplace and enter a proper rational function in s.

What is the region of convergence?

For the usual causal signals, ROC is Re(s) greater than the real part of the rightmost pole.

Results are estimates for informational purposes and are not financial, legal, or professional advice. Last reviewed 2026-08-20.