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Laplace Transform Calculator

Unilateral Laplace table of causal elementary pairs: exp, sin, cos, step, ramp, t^n, damped sin/cos. Evaluate F(s) or f(t). Discovery /inverse-laplace. Not DFT. Not Z. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 10/10 tests · Production surface contract 3/3 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Unilateral Laplace or inverse of a causal elementary pair, evaluated at real s or t.
Scope
Unilateral, causal t≥0
Verification
Engine tested · 10/10 tests · Production surface contract 3/3 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
3 golden · 2 boundary · 5 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

DefinitionF(s) = ∫₀^∞ e^{−s t} f(t) dt
ExponentialL{e^{a t}} = 1/(s−a), Re(s)>a
iTable pairs only. L{e^{−t}} at s=1 is 1/2. L{1} at s=1 is 1. Inverse at t=0 of e^{−t} is 1. Z-transform is a later canonical.

How to use

1

Choose Laplace or inverse, and a pair

Forward needs s in the ROC. Inverse needs t≥0. exp uses a; sin/cos use ω; t^n uses integer n=0…8.

2

Read F(s) or f(t)

The formula and ROC are part of the result. Poles sit on the ROC boundary.

Example calculations

Common configurations with formula and result.

ϟ

L{e^{−t}} at s=1

f=exp, a=−1, s=1

1/(s−a)
0.5
ϟ

L{1} at s=1

f=step, s=1

1/s
1
ϟ

Inverse e^{−t} at t=0

mode=inverse, f=exp, a=−1, t=0

e^{a t}
1

Laplace Transform calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The unilateral Laplace transform is F(s)=∫₀^∞ e^{−st} f(t) dt. This seed evaluates a table of causal elementary pairs at a real s (forward) or t (inverse). Discovery /inverse-laplace. Not a CAS, not Bromwich, not two-sided, not Z, not DFT.
What it calculates
Unilateral Laplace or inverse of a causal elementary pair, evaluated at real s or t.
Inputs
  • mode
  • f
  • s?
  • t?
  • a?
  • omega?
  • n?
Outputs
  • value
  • f
  • roc
  • algorithm
Formula
F(s)=∫₀^∞ e^{−st} f(t) dt (table)
Assumptions
  • Unilateral, causal t≥0
  • Real s in the ROC
  • Table of named pairs, not a CAS
  • Not DFT / FFT
  • Not Z-transform
  • Not Bromwich / two-sided
Units
  • dimensionless
Boundary conditions
  • s not in ROC → INVALID_INPUT
  • t < 0 → INVALID_INPUT
  • n not an integer 0–8 → INVALID_INPUT
  • unknown mode (including dft, z, cas) → INVALID_MODE
Example
mode=laplace f=exp a=-1 s=1 → 0.5
Validation cases

3 published on this page · 10/10 tests · Production surface contract 3/3 · View evidence

  • mode=laplace f=exp a=-1 s=1 → 0.5
  • mode=laplace f=step s=1 → 1
  • mode=inverse f=exp a=-1 t=0 → 1
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

Other calculators in this family: Bessel I Calculator, Bessel J Calculator, Bessel K Calculator, Bessel Y Calculator, Beta Function Calculator, Brent Method Calculator, Definite Integral Calculator, DFT Calculator . Explore all Numerical Calculus.

→ Bessel I Calculator I_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel J Calculator J_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel K Calculator K_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel Y Calculator Y_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Beta Function Calculator Complete Euler B(x, y) = Γ(x)Γ(y)/Γ(x+y) for x>0, y>0. Same engine as Special Functions. Not incomplete beta. Not CAS. Runs locally. › → Brent Method Calculator Brent’s method (golden section plus inverse parabolic interpolation) for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not root-finding Brent. Runs locally. › → Definite Integral Calculator Composite trapezoid, Simpson 1/3, or Gauss–Legendre of a polynomial, sin, or exp on [a, b]. Not indefinite symbolic integrate. Runs locally. › → DFT Calculator Forward DFT X[k]=Σ x[n] e^{−2πi kn/N}, N≤64, unnormalized. Same engine as Discrete Fourier Transform. Radix-2 FFT is on /fft. Not inverse, not Laplace. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a CAS Laplace of an arbitrary expression?

No. This seed is a unilateral table of named causal pairs, evaluated at a real s or t. Not Bromwich. Not symbolic.

Is this DFT or Z?

No. DFT is math.transform.dft. Z-transform is a later canonical. Not CAS.

Where does this run?

Locally in the browser by default. REST and MCP call the same laplace-transform engine.