Inverse Laplace Transform Calculator

Laplace Expression (F(s))

Common Transforms

Calculation Steps

Calculation History

Time	Input (F(s))	Output (f(t))

Powerful tool for converting Laplace-domain functions to time-domain equations. Essential for engineers and mathematicians solving differential equations. Fast, accurate, and mobile-friendly.

Inverse Laplace Transform Calculator Documentation

What Is It?

The Inverse Laplace Transform Calculator is a specialized computational tool that converts complex frequency-domain equations (F(s)) into their corresponding time-domain functions (f(t)). This transformation is crucial in engineering, physics, and control systems analysis where differential equations need to be solved in the time domain.

Core Mathematical Principle

The inverse Laplace transform is defined by the complex integral:

        γ+i∞
f(t) = (1/2πi) ∫ e^(st)F(s) ds
        γ-i∞

Where:

F(s) = Laplace domain function
f(t) = Time domain function
γ = Real number greater than all singularities of F(s)

How to Use

Input Expression: Enter your Laplace-domain function (e.g., “1/(s^2+9)”)
Instant Calculation: Results appear automatically (with 500ms debounce)
Common Transforms: Click any predefined transform for quick reference
Visual History: View your last 10 calculations in both table and chart formats
Step-by-Step Logic: See the calculation methodology in the process log

FAQs

Q: Why use inverse Laplace transforms?
A: They convert complex frequency-domain equations (used for system analysis) back to time-domain functions needed for practical implementation and solution visualization.

Q: What notation should I use for input?
A: Use standard mathematical notation:

Multiplication: * (e.g., 5*s)
Exponents: ^ (e.g., s^2)
Fractions: / (e.g., 1/(s+1))

Q: Can it handle partial fraction decomposition?
A: The calculator automatically applies decomposition rules when matching against its transform database.

Q: Is there mobile support?
A: Yes - the responsive design works perfectly on all devices from smartphones to desktop computers.

Terminology Explained

Laplace Transform (F(s))
An integral transform that converts time-domain functions into complex frequency-domain representations

Time-Domain Function (f(t))
The original function of time that describes a physical system’s behavior

Region of Convergence (ROC)
The set of complex numbers s for which the Laplace transform integral converges

Singularities
Points where F(s) becomes infinite (poles of the function)

Bilateral vs Unilateral
Our calculator uses the unilateral Laplace transform (t ≥ 0) which is standard for physical systems

Common Transform Pairs

Laplace Domain (F(s))	Time Domain (f(t))
1/s	1 (unit step)
1/s^2	t
1/(s-a)	e^(at)
ω/(s^2+ω^2)	sin(ωt)
s/(s^2+ω^2)	cos(ωt)