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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.

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
Inverse unilateral Laplace of a causal elementary pair, evaluated at t≥0, with an engine-linked figure of f(t).
Scope
Mode locked to inverse
Verification
Engine tested · Specification 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
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
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.

ExponentialL^{-1}{1/(s−a)} = e^{a t}
SineL^{-1}{ω/(s²+ω²)} = sin(ω t)
iSame engine as /laplace-transform. Numpy-style numerical DFT is a different capability. Forward Laplace lives on the hub.

How to use

1

Choose a pair and t≥0

exp uses a; sin/cos use ω; t^n uses integer n=0…8. Default e^{−t} at t=0 is 1.

2

Read f(t)

The matching F(s) formula is shown. Poles and ROC belong to the forward transform. The figure is f(t) for t≥0.

Example calculations

Common configurations with formula and result.

ϟ

e^{−t} at t=0

f=exp, a=−1, t=0

e^{a t}
1
ϟ

Ramp at t=3

f=ramp, t=3

t
3

Inverse Laplace calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Inverse of the unilateral Laplace table of causal elementary pairs, evaluated at t≥0. This page is mode=inverse on math.transform.laplace. Not a second engine. Not Bromwich contour inversion. Not Z. Not CAS.
What it calculates
Inverse unilateral Laplace of a causal elementary pair, evaluated at t≥0, with an engine-linked figure of f(t).
Inputs
  • f
  • t
  • a?
  • omega?
  • n?
Outputs
  • value
  • f
  • roc
  • algorithm
Formula
table inverse of F(s)
Assumptions
  • Mode locked to inverse
  • Causal t≥0
  • The Result Card includes an engine-linked figure of f(t) for t≥0. Same table as REST/MCP.
  • Table of named pairs, not Bromwich, not CAS
Units
  • dimensionless
Boundary conditions
  • t < 0 → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
f=exp a=-1 t=0 → 1
Validation cases

2 published on this page · 10/10 tests · Production surface contract 5/5 · View evidence

  • f=exp a=-1 t=0 → 1
  • mode=z → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
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 Z Calculator, Laplace Transform 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 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. › → 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. › → 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 second Laplace engine?

No. It is mode=inverse on math.transform.laplace, the same engine as /calc/math/laplace-transform.

Is this Bromwich inversion?

No. Table inverse of the same causal pairs. Not a contour integral. Not a CAS.

Where does this run?

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