Laplace Transform Calculator
Compute the Laplace transform of standard forms: polynomials, exponentials, trig and steps.
About Laplace Transform Calculator
Computes the Laplace transform F(s) = ∫₀^∞ f(t)e^(-st)dt for standard function forms. Supported transforms: • 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{t·e^(at)} = 1/(s-a)² • L{e^(at)sin(ωt)} = ω/((s-a)²+ω²) • L{e^(at)cos(ωt)} = (s-a)/((s-a)²+ω²) • L{u(t-a)} = e^(-as)/s
Formula
Laplace transform definition
F(s) = ∫₀^∞ f(t)·e^(-st) dt
- F(s)
- Transform in s-domain
- f(t)
- Original time-domain function
- s
- Complex frequency variable
Worked example
With the default values below, Laplace Transform Calculator returns F(s) value of 0.25000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Function typeone of tⁿ (power), e^(at) (exponential), sin(ωt), cos(ωt), t·e^(at), e^(at)·sin(ωt), e^(at)·cos(ωt), Unit step u(t-a)
- tⁿ (power)
- n (power/order)minimum 0
- 2
- a (exponential rate / step delay)
- 1
- ω (angular frequency)
- 1
- Evaluate F(s) at s =
- 2
Results
- F(s) value
- 0.25000000
- L{f(t)} = F(s)
- L{t^2} = 2! / s^3 = 2 / s^3
Step by step
- 1Values used
- 2Laplace transform definition
- 3F(s) value= 0.25000000
- 4L{f(t)} = F(s)= L{t^2} = 2! / s^3 = 2 / s^3
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
What are Laplace transforms used for?
They convert differential equations into algebraic equations in the s-domain, making them much easier to solve. Widely used in control systems and signal processing.
What does the convergence condition s > a mean?
The integral only converges when the real part of s exceeds the growth rate a of the exponential. This defines the region of convergence (ROC).