Inverse Laplace Transform Calculator

Q: What is the mathematical definition of the Inverse Laplace Transform?

The formal definition is f(t) = L^{-1}{F(s)} = (1/2πi) ∫ 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.

Compute the inverse Laplace transform of F(s) to find f(t). Get step-by-step solutions, visualizations, and understand the transformation from frequency domain to time domain.

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))

Embed Inverse Laplace Transform Calculator Widget

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.

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

$ 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.

Additional Resources

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

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

Inverse Laplace Transform Calculator

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About Inverse Laplace Transform Calculator

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?

Additional Resources

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