Inverse Laplace Transform Calculator
Calculate the inverse Laplace transform of mathematical functions instantly. Free online tool for engineering, control systems, and differential equations.
Function Input Details
Enter your s-domain function F(s) to find the time-domain equivalent f(t)
Enter standard rational functions. Use ^ for powers (e.g., s^2). Supported: c/s^n, c/(s±a), s/(s²+a²), c/(s²+a²).
Standard Laplace transforms map s-domain to t-domain.
Time Domain Result
Equivalent function f(t) in the time domain
Enter a valid s-domain function F(s) above, then click Calculate f(t) to find the inverse Laplace transform.
Common Laplace Transform Pairs
Familiarise yourself with these standard transform pairs. They form the foundation of solving differential equations and analysing control systems.
| Time Domain f(t) | s-Domain F(s) | Context / Details |
|---|---|---|
| δ(t) (Dirac Delta) | 1 | Impulse function, used in system response analysis. |
| u(t) (Unit Step) | 1/s | Heaviside step function, represents a signal switching on. |
| t^n | n! / s^(n+1) | Ramp (n=1) or parabolic (n=2) functions. |
| e^(-at) | 1 / (s + a) | Exponential decay, common in RC circuits and thermal systems. |
| sin(ωt) | ω / (s² + ω²) | Sinusoidal oscillation, fundamental in AC circuit analysis. |
| cos(ωt) | s / (s² + ω²) | Cosine oscillation, often appears with initial conditions. |
Laplace Transform FAQ
Everything you need to know about calculating and applying the Inverse Laplace Transform in mathematics and engineering.
The Inverse Laplace Transform is the mathematical operation that converts a function from the complex frequency domain (s-domain) back into the time domain (t-domain). It is the reverse process of the Laplace Transform and is essential for solving linear differential equations and analysing control systems.
Analytically, it is calculated using the complex inversion formula (Brönwich integral). However, in practice, engineers and mathematicians use tables of known Laplace transform pairs and techniques like partial fraction decomposition to match the s-domain function to a known time-domain function.
The Inverse Laplace Transform of 1/s is the Heaviside step function, often denoted as u(t) or H(t). It represents a signal that switches on at t = 0 and remains at 1 for all t > 0.
The Laplace Transform is heavily used in control theory (to analyse system stability via pole-zero plots), electrical engineering (circuit analysis), and signal processing. It converts differential equations into algebraic equations, making them much easier to solve.
This calculator supports standard rational functions commonly found in textbooks. Supported formats include constants (c), powers of s (c/s^n), exponential shifts (c/(s±a)), and sinusoidal forms (s/(s²+a²) and c/(s²+a²)). For complex partial fractions, you may need to decompose them manually first.