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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 Engine tested · Specification checked · 10/10 tests · Production surface contract 3/3 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Inverse unilateral Laplace of a causal elementary pair, evaluated at t≥0.
Scope
Mode locked to inverse
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Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

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.

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.
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
  • 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 3/3 · 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: 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 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.