Inverse Laplace Transform Calculator

Compute the inverse Laplace transform of F(s) to find f(t) with MiniWebtool’s Inverse Laplace Transform Calculator. Follow step-by-step solutions and visualizations to understand the transformation from frequency domain to time domain.

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Help me understand MiniWebtool's Inverse Laplace Transform Calculator: https://miniwebtool.com/inverse-laplace-transform-calculator/. What can I use it for, what inputs or settings do I need, and how do I use it and understand the output? Walk me through one simple example. Use this page as your source and answer in English.

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Split F(s) into partial fractions, then look each one up in the table

Enter F(s) below to get f(t) with the step-by-step solution and the picture of the time-domain function.

Quick Examples - Click to Load

Enter F(s) Function

F(s) =

Use ^ for powers, * for multiplication. Examples: 1/(s^2+4), s/((s-1)*(s+2))

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About Inverse Laplace Transform Calculator

Engineering Mathematics

Frequency Domain

F(s)

Inverse Laplace

Time Domain

f(t)

Welcome to the Inverse Laplace Transform Calculator, a powerful tool for converting functions from the complex frequency domain $ F(s) $ back to the time domain $ f(t) $. Essential for engineers, mathematicians, physicists, and students working with differential equations, control systems, circuit analysis, and signal processing.

At a Glance

Inverse Laplace Transform Calculator converts a function F(s) from the frequency domain to the time-domain function f(t). It shows step-by-step working and graphs of both F(s) and f(t).

Inverse Laplace Transform Calculator: features at a glance
Feature	What you get	Good for
F(s) input	Enter F(s) using s as the variable, with +, -, *, /, ^, parentheses, exp, sin, cos, sqrt, log, pi and E.	Typing transfer functions and transform pairs as written
Supported forms	Supports rational functions, exponentials, and trigonometric functions inside s-domain expressions, and combinations of them.	Differential equations and circuit analysis problems
Time-domain result	Returns f(t) computed with symbolic mathematics.	Finding the closed-form solution of a linear ODE
Step-by-step solution	Shows the working with mathematical explanations.	Following how a result was reached
Two graphs	Plots the time-domain function f(t) and the frequency-domain function F(s).	Comparing a system's two representations
Presets	Common transform presets fill in sample F(s) expressions for quick testing.	Trying the tool before entering your own function

What is the Inverse Laplace Transform?

The Inverse Laplace Transform reverses the Laplace transform operation. Given a function $ F(s) $ in the s-domain (complex frequency domain), it finds the corresponding time-domain function $ f(t) $. This is fundamental for solving linear differential equations with constant coefficients.

Formal Definition

Inverse Laplace Transform

$$f(t) = \mathcal{L}^{-1}\{F(s)\} = \frac{1}{2\pi i} \int_{\gamma - i\infty}^{\gamma + i\infty} e^{st} F(s) \, ds$$

In practice, direct evaluation of this contour integral is rarely performed. Instead, tables of known transform pairs and algebraic manipulation techniques are used to find inverse transforms.

Key Properties

Linearity

$ \mathcal{L}^{-1}\{aF(s) + bG(s)\} = a f(t) + b g(t) $

First Shifting Theorem

$ \mathcal{L}^{-1}\{F(s - a)\} = e^{at} f(t) $

Second Shifting Theorem

$ \mathcal{L}^{-1}\{e^{-as}F(s)\} = u(t-a) f(t-a) $

Convolution Theorem

$ \mathcal{L}^{-1}\{F(s)G(s)\} = \int_0^t f(\tau) g(t - \tau) d\tau $

Common Transform Pairs

Inverse Laplace Transform Calculator: Common Transform Pairs
$ F(s) $	$ f(t) = \mathcal{L}^{-1}\{F(s)\} $
$ \dfrac{1}{s} $	$ 1 $
$ \dfrac{n!}{s^{n+1}} $	$ t^n $
$ \dfrac{1}{s - a} $	$ e^{at} $
$ \dfrac{b}{s^2 + b^2} $	$ \sin(bt) $
$ \dfrac{s}{s^2 + b^2} $	$ \cos(bt) $
$ \dfrac{b}{(s-a)^2 + b^2} $	$ e^{at}\sin(bt) $
$ \dfrac{s-a}{(s-a)^2 + b^2} $	$ e^{at}\cos(bt) $

How to Use This Calculator

Enter F(s): Input your function using standard mathematical notation. Use ^ for exponents, * for multiplication, and standard function names.
Click Calculate: Press the button to compute the inverse Laplace transform using symbolic mathematics.
Review Results: View the time-domain function $ f(t) $, step-by-step solution, and graphical visualizations of both functions.

Applications

Control Systems: Analyze system responses by converting transfer functions to time-domain behavior
Circuit Analysis: Solve RLC circuits and determine transient responses
Signal Processing: Understand filter responses and signal transformations
Differential Equations: Find closed-form solutions to ODEs with constant coefficients
Mechanical Systems: Analyze vibrations, damping, and mechanical responses

Input Syntax Guide

Basic operators: +, -, *, /, ^ (power)
Parentheses: Use ( and ) for grouping
Variable: Use s as the complex frequency variable
Functions: exp(x), sin(x), cos(x), sqrt(x), log(x)
Constants: Use pi for $\pi$ and E for Euler's number

Frequently Asked Questions

What is the Inverse Laplace Transform?

The Inverse Laplace Transform is a mathematical operation that converts a function F(s) from the complex frequency domain (s-domain) back to the time domain f(t). It is the reverse of the Laplace Transform and is essential for solving differential equations in engineering and physics.

How do I use the Inverse Laplace Transform Calculator?

Enter your function F(s) using standard mathematical notation (e.g., 1/(s-7), s/(s^2+4), exp(-2*s)/s). Click Calculate to get the inverse Laplace transform f(t) along with step-by-step solutions and visualizations of both the frequency and time domain functions.

What types of functions are supported?

This calculator supports rational functions (polynomials divided by polynomials), exponential functions, trigonometric functions embedded in s-domain expressions, and combinations thereof. Common forms include 1/(s-a), n!/(s^(n+1)), s/(s^2+b^2), and more complex expressions.

What is the mathematical definition of the Inverse Laplace Transform?

The formal definition is $ f(t) = \mathcal{L}^{-1}\{F(s)\} = \frac{1}{2\pi i} \int_{\gamma - i\infty}^{\gamma + i\infty} e^{st} F(s) \, ds $, where the integral is a contour integral in the complex plane. In practice, tables and algebraic techniques are used rather than direct evaluation of the integral.

Why is the Inverse Laplace Transform important in engineering?

Engineers use the Inverse Laplace Transform to analyze linear time-invariant systems, solve circuit problems, design control systems, and understand signal processing. It converts algebraic equations in the s-domain back to differential equations solutions in the time domain.

Reference this content, page, or tool as:

"Inverse Laplace Transform Calculator" at https://MiniWebtool.com/inverse-laplace-transform-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 24, 2026

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

Chat with your favorite AI

Gemini (copy & open)

For Gemini, paste the copied question into the chat. You can also copy it below.

View the question

Help me understand MiniWebtool's Inverse Laplace Transform Calculator: https://miniwebtool.com/inverse-laplace-transform-calculator/. What can I use it for, what inputs or settings do I need, and how do I use it and understand the output? Walk me through one simple example. Use this page as your source and answer in English.

\( F(s) \)	\( f(t) = \mathcal{L}^{-1}\{F(s)\} \)
\( \dfrac{1}{s} \)	\( 1 \)
\( \dfrac{n!}{s^{n+1}} \)	\( t^n \)
\( \dfrac{1}{s - a} \)	\( e^{at} \)
\( \dfrac{b}{s^2 + b^2} \)	\( \sin(bt) \)
\( \dfrac{s}{s^2 + b^2} \)	\( \cos(bt) \)
\( \dfrac{b}{(s-a)^2 + b^2} \)	\( e^{at}\sin(bt) \)
\( \dfrac{s-a}{(s-a)^2 + b^2} \)	\( e^{at}\cos(bt) \)

Inverse Laplace Transform Calculator

About Inverse Laplace Transform Calculator

At a Glance

What is the Inverse Laplace Transform?

Formal Definition

Key Properties

Common Transform Pairs

How to Use This Calculator

Applications

Input Syntax Guide

Frequently Asked Questions

What is the Inverse Laplace Transform?

How do I use the Inverse Laplace Transform Calculator?

What types of functions are supported?

What is the mathematical definition of the Inverse Laplace Transform?

Why is the Inverse Laplace Transform important in engineering?

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