Inverse Laplace Calculator
Enter any F(s) function and get f(t) with a full step-by-step solution. Includes the complete Laplace transform table with 15+ pairs and a partial fraction decomposition guide.
Enter F(s)
Type your Laplace-domain function. Use s for the variable, ^ for exponents.
Common examples - click to try:
Input format guide:
1/(s+a) → exponential
w/(s^2+w^2) → sine
s/(s^2+w^2) → cosine
1/(s+a)^2 → t·e^(-at)
c/s^n → polynomial in t
Enter a function F(s) to find its inverse Laplace transform f(t)
Instant Pattern Matching
Recognizes exponentials, sinusoids, polynomials, damped oscillations, and direct table matches.
Full Transform Table
15+ Laplace transform pairs organized by category. Click any row to use it in the calculator.
Step-by-Step Solutions
Every result includes the matching table pair, the reasoning, and each algebraic step shown clearly.
Most Common Inverse Laplace Pairs
| F(s) | f(t) = L⁻¹{F(s)} | Category | Notes |
|---|---|---|---|
| 1/s | 1 (unit step) | Basic | Valid for t ≥ 0 |
| 1/s² | t | Basic | Ramp function |
| n!/s^(n+1) | tⁿ | Basic | n = 1, 2, 3, ... |
| 1/(s+a) | e^(-at) | Exponential | Most common pair |
| 1/(s+a)² | te^(-at) | Exponential | Repeated pole |
| ω/(s²+ω²) | sin(ωt) | Trig | Pure sine |
| s/(s²+ω²) | cos(ωt) | Trig | Pure cosine |
| ω/((s+a)²+ω²) | e^(-at)sin(ωt) | Damped | Underdamped system |
| (s+a)/((s+a)²+ω²) | e^(-at)cos(ωt) | Damped | Damped cosine |
| a/(s²-a²) | sinh(at) | Hyperbolic | Unstable system |
| s/(s²-a²) | cosh(at) | Hyperbolic | Hyperbolic cosine |
How Laplace Transforms Work
The Laplace transform is one of the most powerful tools in engineering mathematics. It converts differential equations - which are hard to solve directly - into simple algebraic equations that are easy to manipulate. Once you have the algebraic solution in the s-domain, you use the inverse Laplace transform to convert it back to a time-domain function.
In practice, almost no one computes the Bromwich integral directly. Instead, you use a table of known transform pairs and the properties of linearity, shifting, and scaling. Partial fraction decomposition breaks any rational F(s) into simple fractions that match table entries. This calculator handles the pattern matching and shows you each step of the lookup process.
When You Need It
Frequently Asked Questions
Related Math Tools
This calculator uses pattern matching for common Laplace transform forms. For complex symbolic computation, consider Wolfram Alpha or CAS software. Always verify answers with your course textbook or instructor.