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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 CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 3/3 declared partitions (laplace, inverse, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Unilateral Laplace or inverse of a causal elementary pair, evaluated at real s or t, with an engine-linked figure of F(s) in the ROC or f(t) for t≥0.
Scope
Unilateral, causal t≥0
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.special-transform-cvp
Verification revision
2026-09-22.special-transform-cvp · 1/1 property · digest cc9b93cb51b3
Legacy regression
10/10 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
9/9 golden · 2/2 CVP boundary · 4/4 invalid · 1/1 property · 3/3 cross-interface · 5/5 CVP contract · Manifest
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
  • Oppenheim, Willsky & Nawab, Signals and Systems — unilateral Laplace table
Sources
Evidence
3 legacy golden · 2 legacy boundary · legacy regression suite · 9/9 oracle-backed golden · 4/4 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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. The figure is F(s) on the real ROC, or f(t) for t≥0.

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, with an engine-linked figure of F(s) in the ROC or f(t) for t≥0.
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
  • The Result Card includes an engine-linked figure of F(s) on the real ROC (pole at the ROC boundary) or f(t) for t≥0. Same table as REST/MCP.
  • Not DFT / FFT
  • Not Z-transform
  • Not Bromwich / two-sided
  • Not a pole-zero Argand of complex s
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 5/5 · 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
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Unilateral Laplace transform of a causal elementary pair
  • Oppenheim, Willsky & Nawab, Signals and Systems — unilateral Laplace table
    Supports: L{e^{at}}=1/(s−a); L{1}=1/s; L{t^n}=n!/s^{n+1}; ROC Re(s)>pole
Calculation version
1.0.0

Other calculators in this family: DFT Calculator, Discrete Fourier Transform Calculator, FFT Calculator, Inverse DFT Calculator, Inverse FFT Calculator, Inverse Laplace Calculator, Inverse Z Calculator, Z Transform Calculator . Explore all Transforms.

→ 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. › → Discrete Fourier Transform Calculator Direct DFT / IDFT and radix-2 FFT / IFFT of a length-N sequence, N≤64. FFT requires N=2^k. Numpy backward: unnormalized forward, 1/N inverse. Discovery /dft, /inverse-dft, /fft, /ifft. Not Laplace. Not CAS. Runs locally. › → FFT Calculator Radix-2 Cooley–Tukey FFT, N=2^k ≤64, unnormalized. Same engine as Discrete Fourier Transform. Numpy backward. Not a second DFT engine. Not CAS. Runs locally. › → Inverse DFT Calculator Inverse DFT x[n]=(1/N) Σ X[k] e^{2πi kn/N}, N≤64, Numpy backward. Same engine as Discrete Fourier Transform. Radix-2 FFT is on /fft. Not Laplace. Runs locally. › → Inverse FFT Calculator Radix-2 Cooley–Tukey inverse FFT, N=2^k ≤64, Numpy backward 1/N. Same engine as Discrete Fourier Transform. Not a second DFT engine. Not CAS. Runs locally. › → Inverse Laplace Calculator Inverse of the unilateral Laplace table: exp, sin, cos, step, ramp, t^n, damped sin/cos at t≥0. Same engine as Laplace Transform. Not Bromwich. Not Z. Not CAS. Runs locally. › → Inverse Z Calculator Inverse of the unilateral Z table: a^n, step, ramp, n a^n, sin, cos, damped sin/cos at integer n≥0. Same engine as Z Transform. Not residue. Not Laplace. Not CAS. Runs locally. › → Z Transform Calculator Unilateral Z table of causal elementary pairs: a^n, step, ramp, n a^n, sin, cos, damped sin/cos. Evaluate X(z) or x[n]. Discovery /inverse-z. Not DFT. Not Laplace. Not CAS. Runs locally. ›

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.