Laplace Transform Calculator
This calculator works from the standard transform table and the rules that move it: linearity across a sum, the first shifting theorem for exp(-2*t), the second for a delay, the multiply-by-t rule, and L{f'(t)} = s*F(s) - f(0) for a derivative. Every step is named, and the domain assumption t ≥ 0 and the region of convergence are printed beside the answer rather than left out. So t^2 comes back as 2/s^3 with Re(s) > 0, sin(3*t) as 3/(s^2 + 9), and the inverse of 1/(s + 2) as exp(-2*t) with Re(s) > -2. Every answer is transformed back the other way and matched against what you typed before the page shows it, and a form outside the list it carries is named as unsupported rather than answered with something that merely looks like a result.
f(t) = t^2
F(s) = 2/s^3
from the standard pairs and rules- L{t^n} = n!/s^(n+1)
- L^-1{F(s)} = f(t)
- F(s)
- 2/s^3
- Direction
- Forward
- Domain
- t ≥ 0
- Region
- Re(s) > 0
F(s) = 2/s^3, Re(s) > 0, t ≥ 0
Your expression
Every step, and the rule it used
- Standard pairL{t^n} = n!/s^(n+1)
Power, for the term t^2.
- Checked backL^-1{F(s)} = f(t)
The transform was inverted again and matched your function term for term before it was shown.
Type a function of t and the transform follows, or switch the direction and type an expression in s to go back the other way. Powers, exp, sin, cos, sinh, cosh, the unit step, the Dirac impulse and diff for a derivative are what the tables below list, and anything outside them is named rather than guessed at. Multiplication can be written out or left implicit, so 3t and 3*t both read as three times t, and e^(-2t) reads as exp(-2*t). The answer comes back in the notation this page reads, so it can go straight back into the box in the other direction. Your expression is kept in this browser so the page opens where you left it, and the Start over button at the top of the page forgets it.
These are unilateral transforms: the integral runs from 0 to infinity, so the answer assumes f(t) = 0 for t < 0 and says nothing about negative time.
Answers are printed in the notation this page reads, so an answer can go straight back into the box in the other direction whenever it is inside the size limits below: exp(-2*t), sin(3*t), heaviside(t - 2), delta(t).
Up to 500 characters and powers up to 12. Nothing is cut short: a longer expression is refused with a message rather than quietly trimmed.
The pairs it knows
| f(t) | F(s) | Region | Rule |
|---|---|---|---|
| 1 | 1/s | Re(s) > 0 | Constant |
| t | 1/s^2 | Re(s) > 0 | Power |
| t^2 | 2/s^3 | Re(s) > 0 | Power |
| t^5 | 120/s^6 | Re(s) > 0 | Power |
| exp(-2*t) | 1/(s + 2) | Re(s) > -2 | Exponential |
| t*exp(-2*t) | 1/(s + 2)^2 | Re(s) > -2 | Power times exponential |
| sin(3*t) | 3/(s^2 + 9) | Re(s) > 0 | Sine |
| cos(3*t) | s/(s^2 + 9) | Re(s) > 0 | Cosine |
| sinh(3*t) | 3/(s^2 - 9) | Re(s) > 3 | Hyperbolic sine |
| cosh(3*t) | s/(s^2 - 9) | Re(s) > 3 | Hyperbolic cosine |
| exp(-t)*sin(3*t) | 3/((s + 1)^2 + 9) | Re(s) > -1 | First shift |
| t*sin(3*t) | 6*s/(s^2 + 9)^2 | Re(s) > 0 | Multiply by t |
| t*cos(3*t) | (s^2 - 9)/(s^2 + 9)^2 | Re(s) > 0 | Multiply by t |
| heaviside(t-2) | exp(-2*s)/s | Re(s) > 0 | Second shift |
| heaviside(t-2)*(t-2)^2 | 2*exp(-2*s)/s^3 | Re(s) > 0 | Second shift |
| delta(t) | 1 | all s | Dirac impulse |
| delta(t-2) | exp(-2*s) | all s | Dirac impulse |
| diff(sin(3*t)) | 3*s/(s^2 + 9) | Re(s) > 0 | Derivative |
The rules it uses
| Rule | Identity | Where it is used |
|---|---|---|
| Linearity | L{a*f(t) + b*g(t)} = a*F(s) + b*G(s) | Every sum is split here first, and the transforms are added back over one denominator. |
| Standard pair | L{t^n} = n!/s^(n+1) | The table row each term is looked up in, before any of the rules below move it. |
| First shift | L{exp(a*t)*f(t)} = F(s - a) | An exponential factor shifts the whole transform along s, which is why exp(-2*t)*sin(3*t) lands on (s + 2)^2 + 9. |
| Second shift | L{u(t - c)*f(t - c)} = exp(-c*s)*F(s) | A delay written in t - c comes out as an exponential factor and leaves the region of convergence alone. |
| Multiply by t | L{t*f(t)} = -F'(s) | One factor of t beside a sine or cosine differentiates the transform, which squares the denominator. |
| Derivative | L{f'(t)} = s*F(s) - f(0) | diff(f) is transformed with this rule, and the value of f at 0 is printed beside it. |
| Proper fraction | deg N < deg D | The inverse starts here: a numerator that is not smaller is divided out first, and a constant quotient is an impulse. |
| Factor the denominator | D(s) = product of linear and irreducible quadratic factors | Rational roots are found by search; a factor with none is named as unsupported rather than approximated. |
| Partial fractions | N(s)/D(s) = sum of one fraction per factor power | The coefficients are solved exactly over the rationals and added back up against your numerator. |
| Checked back | L{L^-1{F(s)}} = F(s) | Every answer is run through the other direction and matched against what you typed before it is shown. |
What you can type
| Form | Written as | What it carries |
|---|---|---|
| Powers | t, t^2, t^12 | Whole number powers up to 12. |
| Exponential | exp(-2*t), e^(-2*t) | Any rational rate, positive or negative. |
| Sine and cosine | sin(3*t), cos(3*t) | Any rational frequency. A phase inside the bracket is not read. |
| Hyperbolic | sinh(3*t), cosh(3*t) | Same shape as the trigonometric pair, with s^2 - w^2 underneath. |
| Unit step | heaviside(t - 2), u(t - 2) | A delay, as long as everything beside it is written in t - 2. |
| Impulse | delta(t), delta(t - 2) | The Dirac impulse, on its own or with a number in front. |
| Derivative | diff(sin(3*t)) | d/dt, transformed with s*F(s) - f(0). It nests: diff(diff(f)). |
| Multiply by t | t*sin(3*t) | One factor of t beside a trigonometric or hyperbolic term. |
Common questions
- What does this Laplace transform calculator actually compute?
- The unilateral transform, the one with the integral running from 0 to infinity, so the answer assumes f(t) = 0 for t < 0 and says nothing about negative time. 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 of any rational rate, sin, cos, sinh and cosh of any rational frequency, the unit step written as heaviside(t - 2) or u(t - 2), the Dirac impulse delta(t), 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. 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, and the inverse is the same page with the direction flipped. It divides out an improper fraction first, factors the denominator 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, an irreducible quadratic may be squared, and a factor exp(-2*s) comes back as a delayed step. 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 takes the largest exponential rate in your function and prints the half plane to the right of it, 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.
- 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 a denominator factor with no rational root are all refused by name, and each message says what to do instead. 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.
Forward and inverse unilateral transforms for the elementary pairs this page lists, worked from the standard table with the linearity, shifting, multiply-by-t and derivative rules in exact rational arithmetic, and printed with the domain assumption t ≥ 0 and the region of convergence. Every answer is transformed back the other way and matched against what you typed before it is shown. An expression outside the supported forms is named as unsupported rather than answered, so a plausible-looking result is never printed in place of one the page can stand behind.