Laplace Transform Calculator

What does this laplace transform calculator compute?

The unilateral Laplace transform is F(s)=∫₀^∞ e^{−st} f(t) dt. This calculator evaluates a table of causal elementary pairs at a real s (forward) or t (inverse).

The published formula or method is F(s)=∫₀^∞ e^{−st} f(t) dt (table). The following references support the stated mathematics or method:

• NIST DLMF §1.14 — Integral Transforms: The one-sided Laplace integral and transform/inverse-transform relationships.

Which inputs and limits should you check?

This calculator evaluates its published table of causal one-sided transform pairs. It does not perform unrestricted symbolic transformation or general contour inversion.

The worked example uses f=exp, a=−1, s=1 with 1/(s−a). Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.transform.laplace v1.0.4 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical precision and input policy

These table identities use approximate binary64 evaluation, published to 12 significant decimal digits. The independent reference fixture tolerance is 2e-11 relative for each nonzero scalar, with mathematical zero checked exactly. This is a tested fixture policy, not a uniform error proof. Inputs are dimensionless real parameters; omega is angular frequency in radians per unit time. A real s must lie strictly inside the stated conservative ROC, including degenerate zero-sine pairs. Inverse time must be nonnegative; this table uses u(0)=1 and t^0=1 at t=0.

Finite decimal numbers and decimal strings are accepted. Omitted mode and pair default to laplace and exp. Optional a, omega and n default to 0, 1 and (1 for power, otherwise 0); blank optional form fields use those defaults. The active s or t is required. n is an integer from 0 through 8. Aliases method/mode, pair/integrand/f and w/omega must agree; unknown fields, null, arrays, booleans and hexadecimal values are rejected. Nonzero decimal input that would become zero is rejected as NUMERIC_UNDERFLOW.

Near an ROC boundary, s minus the pole controls sensitivity; high powers amplify it. Trigonometric zeros are sensitive to phase and exponential values to a times t. Input rounding and numerical evaluation error are separate effects. Unrepresentable nonzero outputs, unstable inverse phase products or coarse subnormal quantization produce coded range or precision errors. The curve is a sampled illustration; unavailable samples and unresolved oscillations are reported rather than hidden as zero. The inverse result is f(t); the listed ROC belongs to its forward pair. Host transcendental functions are approximate; cross-runtime fixtures do not certify all inputs.

Sharing and privacy

Interactive calculations run locally in the browser. Share URLs send the included inputs to the server and may appear in logs; REST and MCP also receive their request inputs. Browser calculation history can store results on this device. Log retention, processing region and deletion guarantees are not established by this calculator. Shared version parameters identify the original calculation version; reopening uses the current engine and does not provide a historical execution environment.