Mathematics · Calculus · Integral Transforms

Laplace Transform Calculator

Compute forward and inverse Laplace transforms with complete step-by-step derivations. Supports elementary functions, products, and rational functions via partial fractions.

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Formula

The Laplace transform converts a time-domain function f(t) into a complex frequency-domain function F(s). The transform exists for values of s satisfying Re(s) > σ₀, where σ₀ is the abscissa of convergence.

Laplace Transform Table

30 standard transform pairs with regions of convergence.

Namef(t)F(s)ROC

Delta function

Shifted delta

Unit step

Shifted step

Ramp

Power

General power

Exponential

Sine

Cosine

Hyp. sine

Hyp. cosine

Damped exp.

Damped sine

Damped cosine

t·sin(ωt)

t·cos(ωt)

t·e^(at)

t²·e^(at)

sin²(ωt)

cos²(ωt)

sin·cos

1−cos(ωt)

ωt−sin(ωt)

t·sin(ωt)/2ω

√t

1/√t

erf(√t)

Bessel J₀

ln(t)

Delta function

f(t)

F(s)

ROC:

Shifted delta

f(t)

F(s)

ROC:

Unit step

f(t)

F(s)

ROC:

Shifted step

f(t)

F(s)

ROC:

Ramp

f(t)

F(s)

ROC:

Power

f(t)

F(s)

ROC:

General power

f(t)

F(s)

ROC:

Exponential

f(t)

F(s)

ROC:

Sine

f(t)

F(s)

ROC:

Cosine

f(t)

F(s)

ROC:

Hyp. sine

f(t)

F(s)

ROC:

Hyp. cosine

f(t)

F(s)

ROC:

Damped exp.

f(t)

F(s)

ROC:

Damped sine

f(t)

F(s)

ROC:

Damped cosine

f(t)

F(s)

ROC:

t·sin(ωt)

f(t)

F(s)

ROC:

t·cos(ωt)

f(t)

F(s)

ROC:

t·e^(at)

f(t)

F(s)

ROC:

t²·e^(at)

f(t)

F(s)

ROC:

sin²(ωt)

f(t)

F(s)

ROC:

cos²(ωt)

f(t)

F(s)

ROC:

sin·cos

f(t)

F(s)

ROC:

1−cos(ωt)

f(t)

F(s)

ROC:

ωt−sin(ωt)

f(t)

F(s)

ROC:

t·sin(ωt)/2ω

f(t)

F(s)

ROC:

√t

f(t)

F(s)

ROC:

1/√t

f(t)

F(s)

ROC:

erf(√t)

f(t)

F(s)

ROC:

Bessel J₀

f(t)

F(s)

ROC:

ln(t)

f(t)

F(s)

ROC:

How it works

For the forward transform, the engine identifies the type of input — delta, step, power, exponential, trigonometric, hyperbolic, or products thereof — and applies the corresponding closed-form rule. The first shifting theorem () and multiplication-by-t property () handle compound forms.

For the inverse transform, rational functions are processed via partial fraction decomposition. The denominator polynomial is factored numerically using Newton's method and Bairstow's algorithm for complex conjugate pole pairs. Each partial fraction term is then mapped back to the time domain using the standard inverse table.

Worked example

Find the Laplace transform of .

Apply the first shifting theorem: , then replace s with (s + 2):

, valid for .

Enter e^(-2t)*sin(3t) in the calculator above to see the full derivation.

Frequently asked questions

What functions does this Laplace transform calculator support?

The calculator supports delta functions δ(t), unit step u(t), powers t^n, exponentials e^(at), sine, cosine, hyperbolic sine and cosine, and products of these. For the inverse transform, it handles rational functions F(s) = N(s)/D(s) via partial fraction decomposition.

What is the Region of Convergence (ROC)?

The ROC is the set of complex values s for which the Laplace integral converges. It is typically a right half-plane Re(s) > σ₀, where σ₀ is determined by the growth rate of f(t). Knowing the ROC is essential for correctly inverting the transform.

How does the inverse transform handle complex poles?

Complex conjugate pole pairs are handled by completing the square in the denominator. For a pair at s = α ± jβ, the contribution maps to a damped sinusoid: e^(αt)[A·cos(βt) + B·sin(βt)] in the time domain.

What is the difference between the initial value and final value theorems?

The Initial Value Theorem gives f(0⁺) = lim(s→∞) sF(s) — the behavior right at t=0. The Final Value Theorem gives lim(t→∞) f(t) = lim(s→0) sF(s) — the steady-state value, but only if all poles of sF(s) lie in the left half-plane.

Last updated: 2026-04-04 · Formulas verified against standard references.