Laplace Transform Calculator: A Complete Guide
The Laplace Transform Calculator on this page is a free, browser-based tool that converts a function of time f(t) into a function of complex frequency F(s), and vice versa. Whether you call it Laplace Transformation, the Laplace operator, or simply "the Laplace," this integral transform sits at the heart of linear systems analysis and differential equation solving. Our calculator handles both directions — forward and inverse — with full symbolic step-by-step working and an optional numerical evaluation at any value of s or t.
What Is the Laplace Transform?
Formally, the Laplace transform of a function f(t), defined for t ≥ 0, is given by the improper integral:
The Laplace symbol is typically written as ℒ{} or simply L{}. The variable s is a complex number with sufficiently large real part to make the integral converge. By converting differentiation in time into multiplication by s, the Laplace transform turns calculus problems into algebra — which is far easier to solve. This is why it appears in every undergraduate engineering maths syllabus.
How to Use the Laplace Transform
The most common workflow is: take a differential equation in t, apply the Laplace transform to both sides, solve the resulting algebraic equation for F(s), and then apply the inverse Laplace transform to recover f(t). For example, the rule ℒ{f'(t)} = s·F(s) − f(0) converts a first derivative into a simple multiplication. Higher derivatives follow the same pattern, which is why the technique scales so well to higher-order linear ODEs.
The Laplace Transform Table
Rather than computing the integral every time, engineers use a Laplace Transform table of common pairs. Below is a compact reference of the most-used formulas our calculator applies automatically:
| f(t) | F(s) = ℒ{f(t)} |
|---|---|
1 | 1/s |
t | 1/s² |
t^n (n ∈ ℕ) | n! / s^(n+1) |
e^(at) | 1 / (s − a) |
sin(at) | a / (s² + a²) |
cos(at) | s / (s² + a²) |
sinh(at) | a / (s² − a²) |
cosh(at) | s / (s² − a²) |
t·e^(at) | 1 / (s − a)² |
e^(at)·sin(bt) | b / ((s−a)² + b²) |
e^(at)·cos(bt) | (s−a) / ((s−a)² + b²) |
u(t) (unit step) | 1/s |
δ(t) (impulse) | 1 |
Inverse of Laplace Transform
The inverse Laplace Transform reverses the operation, converting F(s) back into the time-domain function f(t). Mathematically, the inverse is given by the Bromwich integral:
In practice, engineers almost never evaluate this contour integral directly. Instead, they rewrite F(s) using partial fractions, match each piece to a known entry in the Laplace transform table, and sum the results. Our calculator does this matching automatically — for example, it recognises 1/(s−2) as e^(2t), 3/(s²+9) as sin(3t), and s/(s²+4) as cos(2t). For non-standard expressions, the tool falls back to a verified numerical inversion using the Gaver-Stehfest algorithm.
Examples of Laplace Transform
Here are a few worked examples you can try in the calculator right now:
ℒ{sin(2t)} = 2 / (s² + 4)ℒ{e^(−3t)} = 1 / (s + 3)ℒ{t²} = 2 / s³ℒ{t·e^(2t)} = 1 / (s − 2)²ℒ⁻¹{1/(s²+9)} = (1/3)·sin(3t)ℒ⁻¹{(s+1)/(s²+2s+5)} = e^(−t)·cos(2t)
Real-World Usage
Beyond the classroom, the Laplace transform is used to model RLC circuits in electrical engineering, vibration and mass-spring-damper systems in mechanical engineering, process control loops in chemical engineering, and signal flow in communications. Control theorists use it to define transfer functions, while probabilists use a related form (the moment-generating function) to study random variables. Wherever a system is described by linear differential equations, the Laplace transform is usually the cleanest way to solve it.
Tips for Accurate Results
When entering expressions, always use an explicit multiplication symbol (*) between numbers and variables — write 2*t rather than 2t. Use the caret (^) for powers, and remember that e is the constant Euler's number. For piecewise or step functions, use the unit step notation u(t). If the symbolic engine cannot match your input, the calculator still returns a verified numerical value at the point you specify, so you always get a usable answer.
Start by trying one of the example chips below the input field, then experiment with your own functions. Every result is computed locally in your browser — no data is sent anywhere — so you can use this tool with confidence for homework, research, or professional work.