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Laplace Transform Calculator
Enter a function of t and get its Laplace transform F(s), term by term, with the table rule or theorem used for each piece, the region of convergence and a numerical check against the defining integral. Unit steps u(t − c) and impulses δ(t − c) are supported.
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Use t as the variable. Supports powers of t (including √t), e^(at), sin, cos, sinh, cosh, their products and powers, the unit step u(t − c) and the impulse delta(t − c). e^-3t is read as e^(−3t).
Compares F(s) with the defining integral computed numerically.
Graph of f(t)
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How to use the Laplace Transform Calculator
- Type f(t) using t as the variable, for example
t^2*e^(-3t)or(t-2)*u(t-2), or tap an example. - Press Calculate Laplace transform to get F(s).
- Read the rules table to see which table entry or theorem was applied to each term.
- Check the region of convergence: F(s) is valid only for Re(s) to the right of the stated value.
- Copy F(s) or change the check value of s to compare the formula with the integral.
Table of common Laplace transforms
| f(t), t ≥ 0 | F(s) = L{f(t)} | Valid for |
|---|---|---|
| 1 | 1/s | Re(s) > 0 |
| tⁿ (n = 1, 2, 3, …) | n!/sⁿ⁺¹ | Re(s) > 0 |
| tʳ (r > −1) | Γ(r + 1)/sʳ⁺¹ (for √t: √π/(2s3/2)) | Re(s) > 0 |
| eᵃᵗ | 1/(s − a) | Re(s) > a |
| tⁿeᵃᵗ | n!/(s − a)ⁿ⁺¹ | Re(s) > a |
| sin(bt) | b/(s² + b²) | Re(s) > 0 |
| cos(bt) | s/(s² + b²) | Re(s) > 0 |
| sinh(bt) | b/(s² − b²) | Re(s) > |b| |
| cosh(bt) | s/(s² − b²) | Re(s) > |b| |
| eᵃᵗsin(bt) | b/((s − a)² + b²) | Re(s) > a |
| eᵃᵗcos(bt) | (s − a)/((s − a)² + b²) | Re(s) > a |
| t·sin(bt) | 2bs/(s² + b²)² | Re(s) > 0 |
| t·cos(bt) | (s² − b²)/(s² + b²)² | Re(s) > 0 |
| u(t − c), c ≥ 0 | e⁻ᶜˢ/s | Re(s) > 0 |
| δ(t − c), c ≥ 0 | e⁻ᶜˢ | all s |
Properties used by the calculator
| Property | Time domain | s-domain |
|---|---|---|
| Linearity | a·f(t) + b·g(t) | a·F(s) + b·G(s) |
| First shifting (frequency shift) | eᵃᵗf(t) | F(s − a) |
| Second shifting (time shift) | f(t − c)·u(t − c) | e⁻ᶜˢF(s) |
| Multiplication by tⁿ | tⁿf(t) | (−1)ⁿF⁽ⁿ⁾(s) |
| Derivative | f′(t) | sF(s) − f(0) |
| Second derivative | f″(t) | s²F(s) − s·f(0) − f′(0) |
| Integral | ∫₀ᵗ f(τ) dτ | F(s)/s |
Tips
- Piecewise functions are sums of steps: a pulse of height 1 from t = 1 to t = 3 is u(t − 1) − u(t − 3), with transform (e⁻ˢ − e⁻³ˢ)/s.
- When the shifted function is not written as g(t − c), for example t²·u(t − 1), the calculator rewrites it as g(t + c) first: (t + 1)² = t² + 2t + 1, giving e⁻ˢ(2/s³ + 2/s² + 1/s).
- Powers of sines and cosines are reduced first: sin²t = 1/2 − cos(2t)/2, so L{sin²t} = 1/(2s) − s/(2(s² + 4)) = 2/(s(s² + 4)).
- The region of convergence is set by the fastest-growing exponential in f(t); polynomial factors and sines do not change it.
Frequently asked questions
What is the Laplace transform?
The Laplace transform turns a function of time f(t), defined for t ≥ 0, into a function of a complex variable s: F(s) = ∫₀^∞ e^(−st) f(t) dt. It converts derivatives into multiplication by s, which is why it is used to solve linear differential equations and to analyze circuits and control systems.
What is the region of convergence?
The defining integral only converges for values of s with a large enough real part. If |f(t)| ≤ M·e^(σt), the transform exists for Re(s) > σ. For example L{e^(3t)} = 1/(s − 3) is valid for Re(s) > 3, and L{sin t} = 1/(s² + 1) for Re(s) > 0.
What is the first shifting theorem?
Multiplying by an exponential shifts the transform: L{e^(at) f(t)} = F(s − a). So since L{cos 2t} = s/(s² + 4), L{e^(−t) cos 2t} = (s + 1)/((s + 1)² + 4).
How do I take the Laplace transform of a unit step function?
Use the second shifting theorem: L{g(t)·u(t − c)} = e^(−cs)·L{g(t + c)}. A plain step u(t − c) gives e^(−cs)/s, and a ramp that starts at t = 2, (t − 2)u(t − 2), gives e^(−2s)/s². Type u(t-2), heaviside(t-2) or step(t-2).
What is the Laplace transform of the Dirac delta?
L{δ(t − c)} = e^(−cs) for c ≥ 0, and L{δ(t)} = 1. By the sifting property g(t)·δ(t − c) transforms to g(c)·e^(−cs).
Why does t·sin(2t) give a squared denominator?
Multiplying by t corresponds to differentiating in s: L{t f(t)} = −F′(s). Differentiating 2/(s² + 4) gives 4s/(s² + 4)², so repeated roots in the denominator of F(s) always come from factors of t in f(t).
Which functions are not supported?
Functions without a closed-form table entry, such as ln t, tan t or 1/(t + 1), and functions like 1/t or e^(t²) whose transform does not exist, are reported instead of guessed. Inverse transforms are not included; for solving ODEs use the differential equation solver.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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