Inverse Laplace transform calculator
Type a fraction of two polynomials in s, such as (s + 3) / ((s + 1)(s + 2)), and get f(t), with the poles and the steps of the partial fraction decomposition.
A ratio of two polynomials in s, such as 1/(s^2 + 4) or 5/(s − 2)^2.
Fill in the fields; the answer appears here straight away.
How it's worked out
How it works
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Enter the values
Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.
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Instant answer
The answer appears as you type, with the most important intermediate values.
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See the working
Under "How it's worked out" you see the steps, handy for checking your own calculation.
How it works
- Find the poles: the zeros of the denominator.
- Split F(s) into partial fractions, one term per pole p, with (s − p)ᵏ in the denominator for repeated poles.
- Transform every term back with the table: L⁻¹{1 / (s − p)ᵏ} = tᵏ⁻¹ / (k − 1)! · e^(pt).
Two complex poles a ± bi together give a term e^(at)·(… cos(bt) + … sin(bt)). For the example, f(t) = 2e^(−t) − e^(−2t).
Limits
The degree of the numerator must be lower than that of the denominator (a proper fraction); otherwise f(t) would contain a delta function. The forward direction is the Laplace transform calculator.
Frequently asked questions
Is the inverse Laplace transform free?
Yes. The inverse Laplace transform is completely free, with no account, sign-up or usage limit.
How accurate is the result?
The tool calculates with your browser's number precision. For important calculations, always double-check what you entered.
Does it work on my phone?
Yes. The tool works in any modern browser on a phone, tablet or computer.