Laplace Transform Calculator

Common questions

What does this Laplace transform calculator actually compute?

The unilateral Laplace integral runs from 0 to infinity, so the answer assumes f(t) = 0 for t < 0 and says nothing about negative time. Supported Laplace integrals are evaluated by exact table and rule algebra. There is no separate operation for the transform of an integral of f, or for finding an antiderivative. It runs both directions from the same table: forward from a function of t, and inverse from an expression in s. Coefficients, rates and frequencies are held as exact fractions, so 0.1*t comes back as 1/(10*s^2) rather than as a rounded decimal.

Which functions can it transform?

Whole number powers up to t^12, exp(a*t) for a rational rate a, sin, cos, sinh and cosh of any rational frequency, the unit step written as heaviside(t - 2) or u(t - 2), standalone Dirac impulses such as delta(t), or rational scalar multiples, diff for a derivative, and sums and products of those within the limits the tables name. One factor of t may sit beside a sine or cosine, which is the multiply-by-t rule, so t*sin(3*t) gives 6*s/(s^2 + 9)^2. Periodic sine and cosine inputs and their sums are supported within these rational-frequency and product limits; arbitrary one-period waveform extension is not available. The tables under the tool list every pair it carries and every rule it uses, and each transform in the pairs table is worked by the same code that answers your expression.

Does it do inverse Laplace transforms?

Yes. A constant such as 2 inverts directly to 2*delta(t); exp(-2*s) moves an impulse to t = 2. For a nonconstant rational denominator, the page divides out an improper fraction first, factors by rational root search into linear and irreducible quadratic pieces, solves the partial fraction coefficients exactly over the rationals, and reads each piece back through the table. Linear factors may repeat, and an irreducible quadratic may be squared. An exponential factor delays the time-domain answer. Denominators up to degree 12 are attempted.

Why does it print a region of convergence?

Because a transform without one is an incomplete answer: 1/(s - 2) is the transform of exp(2*t) only where Re(s) > 2, and the same algebra with a different region describes a different function. The page combines the tail after the last delay, including hyperbolic cancellation, and prints the half plane to the right of its surviving exponential rate, so exp(-2*t) gives Re(s) > -2, sinh(3*t) gives Re(s) > 3, and an impulse on its own converges for all s. A polynomial that stops at a finite breakpoint converges for all s; apparent poles in its printed formula are removable limits.

Why does it say my expression is unsupported?

Because it would rather name the gap than hand back something that looks like an answer. A product of two sines, a phase inside the bracket as in sin(3*t + 1), t^2 beside a sine, and an unsplittable cubic denominator factor are all refused by name, and each message says what to do instead. Nonlinear arguments such as exp(t*exp(t)) or sin(exp(t)-1) are outside the table. Products of two impulses, including delta(t)^2, have no pair in the ordinary distribution table used here. A unit step multiplied by an impulse at its jump is also refused: this page supplies no convention for that boundary product. Derivatives with a step boundary at or after the origin are refused before shifting or simplification can lose the jump; differentiate a smooth function first, then apply an external delay. Two cases worth knowing: 1/(s^2 + 2) needs the frequency sqrt(2), which is not rational, and a cubic such as s^3 + s + 1 is not split for you, though typing the denominator in factored form makes the page use your factors.

How long an expression can it read, and does anything I type leave the browser?

Up to 500 characters, with powers up to 12 and denominators up to degree 12. Nothing is ever trimmed to fit: a longer paste is refused with a message that names the limit, so you never get a quietly different answer. Everything runs in this browser, nothing is uploaded and no account is asked for. Your expression is kept locally so the page opens where you left it, and the Start over button at the top of the page forgets it.

How do I transform a piecewise function?

Open Build a two-branch piecewise function. Enter the first branch for 0 ≤ t < c, a nonnegative decimal or fractional breakpoint c, and the second branch for t ≥ c, using the selected time variable. Apply piecewise function builds and checks the complete function and selects Forward. For 0 before t = 2 and 1 afterward, the answer is exp(-2*s)/s. Constant and supported polynomial branches work; forms needing unsupported shifted phases or other rules are refused, with the previous expression kept. The builder accepts two intervals, not an arbitrary branch table, ODE or initial-value problem.

Which forward and inverse examples can I try?

In Forward, t^2 gives 2/s^3 and sin(3*t) gives 3/(s^2 + 9). In Inverse, 1/(s + 2) gives exp(-2*t). These are unilateral examples with t ≥ 0; the first two have Re(s) > 0, and the inverse example has Re(s) > -2. The page shows their named rules and checks each result in the other direction.