Inverse Laplace Transform Calculator
Find the time-domain function from a standard-form Laplace transform F(s).
About Inverse Laplace Transform Calculator
Finds the inverse Laplace transform f(t) = L⁻¹{F(s)} for standard forms. Supported inverse transforms: • L⁻¹{n!/s^(n+1)} = tⁿ • L⁻¹{1/(s-a)} = e^(at) • L⁻¹{ω/(s²+ω²)} = sin(ωt) • L⁻¹{s/(s²+ω²)} = cos(ωt) • L⁻¹{1/(s-a)²} = t·e^(at) • L⁻¹{ω/((s-a)²+ω²)} = e^(at)sin(ωt) • L⁻¹{(s-a)/((s-a)²+ω²)} = e^(at)cos(ωt)
Formula
Inverse Laplace
f(t) = L⁻¹{F(s)}
- f(t)
- Time-domain function
- F(s)
- S-domain transform
Worked example
With the default values below, Inverse Laplace Transform Calculator returns f(t) value of 0.36787944. Change any input above and every figure here is recalculated in your browser.
Inputs used
- F(s) formone of n!/s^(n+1) → tⁿ, 1/(s-a) → e^(at), ω/(s²+ω²) → sin(ωt), s/(s²+ω²) → cos(ωt), 1/(s-a)² → t·e^(at), ω/((s-a)²+ω²) → e^(at)sin(ωt), (s-a)/((s-a)²+ω²) → e^(at)cos(ωt)
- 1/(s-a) → e^(at)
- n (power)minimum 0
- 2
- a (exponential rate)
- -1
- ω (angular frequency)
- 2
- Evaluate f(t) at t =
- 1
Results
- f(t) value
- 0.36787944
- f(t) =
- f(t) = e^(-1t)
Step by step
- 1Values used
- 2Inverse Laplace
- 3f(t) value= 0.36787944
- 4f(t) == f(t) = e^(-1t)
Assumptions and limits
- The result is derived only from the 5 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax or fees or charges into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
How do I handle more complex F(s)?
Use partial fraction decomposition to break F(s) into the standard forms listed above, then invert each term separately.
What about the e^(-as) factor?
A factor e^(-as) in F(s) corresponds to a time delay: L⁻¹{e^(-as)F(s)} = f(t-a)·u(t-a).