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Laplace transform calculator

Type a function of t, such as 3 + 2t^2 − 4exp(−2t) + 5sin(3t), and get its Laplace transform F(s), term by term from the standard table.

Sums of constants, tⁿ, exp(at), sin(bt), cos(bt), sinh(bt) and cosh(bt), also multiplied by exp(at).

Fill in the fields; the answer appears here straight away.

How it works

  1. Enter the values

    Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.

  2. Instant answer

    The answer appears as you type, with the most important intermediate values.

  3. See the working

    Under "How it's worked out" you see the steps, handy for checking your own calculation.

Supported terms

Constants, tⁿ, e^(at), sin(bt), cos(bt), sinh(bt) and cosh(bt), and those terms multiplied by e^(at) (the shift rule). Because the Laplace transform is linear, sums and constant multiples work too.

Examples from the table

  • L{1} = 1/s
  • L{tⁿ} = n! / sⁿ⁺¹
  • L{e^(at)} = 1 / (s − a)
  • L{sin(bt)} = b / (s² + b²)
  • L{e^(at)·cos(bt)} = (s − a) / ((s − a)² + b²)

The result is valid for s greater than the real part of every a. To go back from F(s) to f(t), use the inverse Laplace transform calculator.

The default example, term by term

3 + 2t² − 4e^(−2t) + 5 sin(3t) becomes 3/s + 4/s³ − 4/(s + 2) + 15/(s² + 9). Each term uses a table entry: L{1} = 1/s, L{t²} = 2/s³, L{e^(−2t)} = 1/(s + 2) and L{sin 3t} = 3/(s² + 9). The constant factors 3, 2, −4 and 5 just carry along.

Solving a differential equation

For y′ + 2y = 0 with y(0) = 1, transform both sides using L{y′} = sY − y(0): sY − 1 + 2Y = 0, so Y = 1/(s + 2). The inverse transform gives y = e^(−2t), a decay that you can also explore with the exponential decay calculator. Use the inverse calculator for the last step in harder cases.

Why engineers use it

In electronics and control, a circuit or system becomes a fraction in s, and cascading two systems means multiplying fractions. Poles of that fraction, the zeros of its denominator, say whether the system is stable: poles with a negative real part give responses that die out.

Frequently asked questions

What does the variable s mean?

It is a complex frequency. The Laplace transform turns a function of time f(t) into a function of s, in which differentiation becomes multiplication by s and many differential equations become algebra.

Which functions are supported?

Constants, tⁿ, e^(at), sin(bt), cos(bt), sinh(bt) and cosh(bt), also multiplied by e^(at), and sums and multiples of those. Products such as t · sin(t) and functions such as 1/t are not in the table.

What does "valid for s greater than …" mean?

The integral that defines the transform only converges for large enough s. For e^(at) you need s > a, for example s > −2 for e^(−2t).

How do I get back to f(t)?

With the [inverse Laplace transform calculator](/en/math/inverse-laplace-transform-calculator), which finds f(t) from a fraction in s.