What is the Laplace transform?

The Laplace transform of a function f(t) is F(s) = integral from 0 to infinity of e^(-st) * f(t) dt. It converts differential equations into algebraic equations in the complex variable s, making them easier to solve.

What is the region of convergence (ROC)?

The ROC is the set of values of s for which the Laplace integral converges. For example, L{e^(at)} = 1/(s-a) is valid only for Re(s) > a. The ROC determines whether the transform exists for a given input.

How do I use this table for inverse Laplace transforms?

Read the table in reverse: if you have a known F(s) expression in the right column, find the corresponding f(t) in the left column. Partial fraction decomposition often helps break complex F(s) into simpler pairs.

What is the unit step function u(t)?

The unit step function u(t) equals 0 for t = 0. In Laplace analysis, it is used to model signals that switch on at t = 0, and its transform is 1/s.

What does the time-shifting theorem say?

If L{f(t)} = F(s), then L{f(t-a)u(t-a)} = e^(-as) * F(s) for a >= 0. This means a time delay of a seconds multiplies the transform by e^(-as).

How does the derivative property work?

L{f'(t)} = s*F(s) - f(0). For higher derivatives: L{f''(t)} = s^2*F(s) - s*f(0) - f'(0). This is what lets you convert differential equations into algebraic ones.

Laplace Transform Calculator

Choose a common function, enter its parameters, and get its one-sided Laplace transform and region of convergence; then browse or search the reference table.

Parameterized transform calculator

Build an exact one-sided transform for a supported causal function. This is not a general computer-algebra solver.

Function c

Time domain

Laplace transform

Region of convergence

#	Name	f(t) — time domain	F(s) — s-domain	ROC	Notes

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Summary

Choose a common function, enter its parameters, and get its one-sided Laplace transform and region of convergence; then browse or search the reference table.

How it works

Choose a supported function and enter its parameter values to build its transform.
Read the substituted time-domain expression, F(s), and region of convergence.
Select a category tab or type a function name in the reference search box.
The table filters to show matching transform pairs.
Each row shows f(t), F(s), the region of convergence, and notes.
Click the copy icon on any row to copy the F(s) formula.
Use the "All" tab to browse the full table of transform pairs.

Use cases

Solving ordinary differential equations using the Laplace method.
Analyzing control system transfer functions and stability.
Circuit analysis in electrical engineering (RLC circuits).
Signal processing and system response calculations.
Verifying hand-calculated transforms during coursework.
Quick reference while studying for engineering exams.
Finding inverse Laplace transforms by reverse lookup.
Identifying the correct transform for piecewise or delayed functions.

Frequently Asked Questions

Last updated: 2026-09-22 · Reviewed by Nham Vu