Inverse Laplace Transform Calculator
Opens the Laplace calculator in inverse mode, converting an expression in the frequency variable back to a time-domain function where the supported symbolic rules allow it.
F(s) = 1/(s + 2)
f(t) = exp(-2*t)
from the standard pairs and rules- deg N < deg D
- D(s) = product of linear and irreducible quadratic factors
- N(s)/D(s) = Q + R(s)/D(s)
- f(t)
- exp(-2*t)
- Direction
- Inverse
- Domain
- t ≥ 0
- Region
- Re(s) > -2
f(t) = exp(-2*t), Re(s) > -2, t ≥ 0
Your expression
Build a two-branch piecewise function
Use the selected time variable in both branches. The first applies from zero to the breakpoint; the second applies from the breakpoint onward. Apply checks the complete function and switches to Forward. Builder fields stay in this visit; the applied expression follows the normal expression save settings.
Every step, and the rule it used
- Proper fractiondeg N < deg D
The numerator is degree 0 against a denominator of degree 1, so it is already a proper fraction and splits straight away.
- Factor the denominatorD(s) = product of linear and irreducible quadratic factors
Rational root search splits higher-degree factors into the supported linear and quadratic pieces. A higher-degree factor with no rational root is named rather than guessed at.
- Partial fractionsN(s)/D(s) = Q + R(s)/D(s)
Q = 0. The proper fractions come from an exact rational coefficient system and recombine to the original numerator. Including Q, the complete decomposition is checked against the original numerator.
- Standard pairL{t^(k-1)*exp(a*t)/(k-1)!} = 1/(s - a)^k
Exponential, read back from the table.
- Checked backL{f(t)} = F(s)
Every term was transformed forward again and matched against the parsed frequency expression before it was shown. This is an algebraic check of the supported forms.
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 root search splits higher-degree factors; a higher-degree factor with no rational root is named as unsupported rather than approximated. Supported irreducible quadratics are retained, with rational frequencies required for their inverses. |
| Partial fractions | N(s)/D(s) = Q + R(s)/D(s) | After division, Q is the constant quotient (zero for a proper fraction). The proper fractions recombine exactly to R/D; including Q checks the complete decomposition against the original numerator. |
| Checked back | L{L^-1{F(s)}} = F(s) | Every answer is run through the other direction and matched against the parsed expression before it is shown. This algebraic check covers the supported forms. |
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 an inverse Laplace transform do?
- It maps a function of the transform variable back to its time-domain function, such as 1/s² to t. The page shows the parsed expression and the resulting form.
- Are all expressions supported?
- No. The calculator names unsupported syntax or an unevaluated result instead of presenting a guess as an answer.
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 supported answer is transformed back the other way and compared with the parsed expression before it is shown. This algebraic check covers the listed forms; unsupported expressions are refused by name. For a finite interval, all s includes removable values at apparent poles of the printed formula.