What is the inverse Laplace transform?

The inverse Laplace transform converts a function F(s) in the complex frequency domain back to the time-domain function f(t). It is used extensively in control systems and circuit analysis to find transient responses.

Does this tool handle all F(s) expressions?

This tool uses a reference table of the most common standard forms. For arbitrary or composite expressions you may need to decompose them into partial fractions first, then look up each term separately.

What do the parameters a, b, n, and omega represent?

They are symbolic constants used in the table entries. "a" and "b" are real constants (e.g., decay rates), "n" is a non-negative integer, and "omega" (w) is angular frequency in radians per second.

What is the region of convergence assumed?

All results assume the causal (right-sided) convention where f(t) = 0 for t < 0, and u(t) denotes the unit step function. This is the standard engineering convention.

Can I use this for partial fraction decomposition?

Yes. After decomposing F(s) into simpler terms, look up each term individually in the table and sum the corresponding f(t) results using the linearity property of the Laplace transform.

Is the result exact or approximate?

Results are exact closed-form expressions from the standard Laplace transform table, not numerical approximations.

Inverse Laplace Calculator

Name: Inverse Laplace Calculator
Availability: InStock
Author: Nham Vu

Select or type a Laplace-domain expression F(s) and instantly get the time-domain inverse f(t) from a built-in reference table.

Transform Table

F(s)	f(t) (t ≥ 0)

Click any row to select it.

Selected Transform

Click a row in the table to see the inverse transform.

Key Properties

Linearity: L^-1{aF+bG} = af(t)+bg(t)
s-shift: L^-1{F(s-a)} = e^atf(t)
t-shift: L^-1{e^-asF(s)} = f(t-a)u(t-a)
Scaling: L^-1{F(s/a)}/a = f(at)
Convolution: L^-1{F(s)G(s)} = (f*g)(t)

Copied!

Summary

Select or type a Laplace-domain expression F(s) and instantly get the time-domain inverse f(t) from a built-in reference table.

How it works

Browse the table or type a search term in the filter box.
Click any row to load that F(s) into the input panel.
Adjust the parameter values (a, b, n, omega) as needed.
The tool matches your expression to the closest table entry and displays f(t).
Copy the result or the LaTeX source with one click.

Use cases

Solve inverse Laplace transforms for control engineering homework.
Verify hand-computed results against the standard table.
Look up the time-domain response of a transfer function.
Find the impulse or step response of an RLC circuit.
Convert frequency-domain expressions back to time domain.
Study common Laplace pairs side-by-side in one reference.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu