What standard function forms are supported?

The calculator supports 14 standard pairs: constant/unit step 1, t, t^n, e^(at), t^n·e^(at), sin(kt), cos(kt), e^(at)·sin(kt), e^(at)·cos(kt), sinh(kt), cosh(kt), t·sin(kt), t·cos(kt), and the Dirac delta δ(t).

What happens if s₀ falls outside the region of convergence?

The tool validates s₀ against the theoretical region of convergence (such as s > a for exponential terms) and flags inputs that do not converge.

Can parameter n take fractional values?

The parameter n is restricted to integers from 1 to 10 to evaluate standard factorial formulations n!/(s^(n+1)).

Why does frequency parameter k require a nonzero value?

Trigonometric and hyperbolic forms rely on a nonzero angular frequency k to prevent degenerate zero expressions or invalid division by zero.

Does the calculator include operational properties in the output?

Yes, the tool displays derivation notes summarizing key properties used, such as the first frequency shift theorem or integration by parts.

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Laplace Transform Calculator (Table Lookup)

Table lookup of L{f(t)} for 14 standard pairs, with parameter substitution, region of convergence, derivation note, and optional numeric F(s₀) evaluation.

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2 examples

L{e^t} = 1/(s − 1), ROC s > 1, and F(2) = 1

The exponential pair: ∫₀^∞ e^(-st) e^t dt = ∫₀^∞ e^(-(s−1)t) dt = 1/(s−1), converging only for s > 1. Evaluating at s = 2 gives exactly 1.

Laplace transform (table lookup) f(t) = e^(t) F(s) = 1/(s - 1) Region of convergence: s > a (here s > 1) Derivation: ∫₀^∞ e^(-(s-a)t) dt = 1/(s - a), the first-shift theorem with e^(at). Evaluation: F(2) = 1

L{sin(2t)} = 2/(s² + 4) — converges for all s

The sine pair: two integrations by parts give k/(s² + k²); with k = 2 the transform is 2/(s² + 4). Since sin is bounded, the integral converges for every s.

Laplace transform (table lookup) f(t) = sin(2t) F(s) = 2/(s^2 + 4) Region of convergence: all s (sin is bounded) Derivation: two integrations by parts (or Euler's formula on e^(ikt)). Evaluation: F(3) = 0.1538

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6 options

SettingsAdjust formats, ranges, numbers, and modes.6

f(t) formselectRequiredParameter a (exponent)numberOptionalExponential rate in e^(at) forms.Parameter k (frequency)numberOptionalAngular frequency in sin/cos/sinh/cosh forms (must be nonzero).Parameter n (power, 1–10)numberOptionalPower in t^n forms (integer 1–10; the factorial n! is used).Evaluate F(s) at s (optional)numberOptionalNumeric evaluation point; must satisfy the region of convergence.Decimal PlacesnumberOptional

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Laplace Transform Calculator (Table Lookup)

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1. Evaluating Exponential Decay in the Laplace Domain

2. Damped Harmonic Oscillator Transform

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