Inverse Laplace Transform Calculator — Step-by-Step Solutions
Compute the inverse Laplace transform of rational functions with detailed steps. Free online inverse Laplace transform calculator using partial fraction decomposition and standard transform tables. Get f(t) from F(s) instantly.
Inverse Laplace Transform Calculator
Enter a rational function F(s) to find its inverse Laplace transform f(t). Supports partial fraction decomposition and common transform pairs.
Inverse Laplace Transform Formula & Definition
The inverse Laplace transform converts a function F(s) from the complex frequency domain (s-domain) back to the time domain f(t). It is denoted as:
The formal definition uses the Bromwich integral (complex inversion formula):
In practice, most inverse Laplace transforms are computed using partial fraction decomposition combined with a table of standard transform pairs, avoiding the need for complex contour integration.
Key Variables
- F(s) — Laplace-domain function (function of complex variable s)
- f(t) — Time-domain function (function of real variable t, defined for t ≥ 0)
- L⁻¹ — Inverse Laplace transform operator
- γ — Real number greater than the real part of all singularities of F(s)
How to Calculate the Inverse Laplace Transform
Follow these steps to compute an inverse Laplace transform using the partial fraction method:
- Check if F(s) is a proper rational function — The degree of the numerator must be less than the degree of the denominator. If not, perform polynomial long division first.
- Factor the denominator — Find all roots (real and complex) of the denominator polynomial.
- Set up partial fractions — For each linear factor (s+a), use A/(s+a). For repeated factors (s+a)ⁿ, use A₁/(s+a) + A₂/(s+a)² + ... + Aₙ/(s+a)ⁿ. For irreducible quadratic factors (s²+bs+c), use (As+B)/(s²+bs+c).
- Solve for coefficients — Multiply both sides by the denominator and solve the resulting system of equations.
- Apply the inverse Laplace transform to each term — Use a standard transform table to convert each partial fraction term from the s-domain to the time domain.
- Sum all time-domain terms — Add all the individual f(t) components to obtain the final result.
Common Inverse Laplace Transform Pairs Table
Reference these standard transform pairs when computing inverse Laplace transforms:
| F(s) | f(t) = L⁻¹{F(s)} | Condition |
|---|---|---|
| 1/s | 1 (unit step u(t)) | t ≥ 0 |
| 1/s² | t | t ≥ 0 |
| 1/sⁿ | tⁿ⁻¹/(n-1)! | n = 1,2,3,... |
| 1/(s+a) | e-at | Any a |
| 1/(s+a)ⁿ | tⁿ⁻¹e-at/(n-1)! | n = 1,2,3,... |
| 1/(s²+a²) | sin(at)/a | a ≠ 0 |
| s/(s²+a²) | cos(at) | a ≠ 0 |
| 1/(s²-a²) | sinh(at)/a | a ≠ 0 |
| s/(s²-a²) | cosh(at) | a ≠ 0 |
| 1/((s+a)²+b²) | e-atsin(bt)/b | b ≠ 0 |
| (s+a)/((s+a)²+b²) | e-atcos(bt) | b ≠ 0 |
| e-as/s | u(t-a) | a ≥ 0 |
Inverse Laplace Transform Worked Examples
Example 1: Simple Exponential
Find L⁻¹{1/(s+3)}
f(t) = e-3t for t ≥ 0
Example 2: Sine Function
Find L⁻¹{1/(s²+4)}
f(t) = sin(2t)/2 for t ≥ 0
Example 3: Partial Fractions Required
Find L⁻¹{1/(s²+3s+2)}
1/((s+1)(s+2)) = 1/(s+1) - 1/(s+2)
f(t) = e-t - e-2t for t ≥ 0
Example 4: Damped Oscillation
Find L⁻¹{(s+1)/(s²+4s+13)}
F(s) = (s+2-1)/((s+2)²+3²) = (s+2)/((s+2)²+9) - 1/((s+2)²+9)
f(t) = e-2tcos(3t) - (1/3)e-2tsin(3t)
People Also Ask About Inverse Laplace Transforms
Frequently Asked Questions About Inverse Laplace Transforms
Inverse Laplace Transform Glossary
Laplace Transform
An integral transform that converts a time-domain function f(t) into a complex frequency-domain function F(s) = ∫₀∞ e⁻ˢᵗf(t)dt.
Inverse Laplace Transform
The operation L⁻¹ that recovers f(t) from F(s), denoted f(t) = L⁻¹{F(s)}.
s-Domain
The complex frequency domain where s = σ + jω. Laplace transforms map time functions into this domain for algebraic manipulation.
Partial Fraction Decomposition
A technique for breaking a complex rational function into a sum of simpler fractions, each of which has a known inverse Laplace transform.
Unit Step Function
Denoted u(t), equals 0 for t < 0 and 1 for t ≥ 0. Its Laplace transform is 1/s, making it fundamental to causal system analysis.
Convolution Theorem
States that L⁻¹{F(s)G(s)} = (f * g)(t) = ∫₀ᵗ f(τ)g(t-τ)dτ, allowing inverse transforms of products via convolution integrals.
Bromwich Integral
The formal complex inversion formula for the inverse Laplace transform: f(t) = (1/2πi)∫ eˢᵗF(s)ds along a vertical contour in the complex plane.
Heaviside Expansion
A formula for the inverse Laplace transform of rational functions with distinct poles, expressing f(t) as a sum of exponential terms using residues.
Editorial Review & Methodology
This inverse Laplace transform calculator was built and reviewed by the NumbrWiz Editorial Team. The inverse Laplace transform is a cornerstone of advanced engineering mathematics, verified against standard textbooks including Oppenheim & Willsky, Kreyszig's Advanced Engineering Mathematics, and standard differential equations curricula.
- Transform pair verification: All standard pairs cross-checked against multiple authoritative sources in signals and systems literature.
- Partial fraction algorithm: Tested with real distinct roots, repeated roots, and complex conjugate pairs.
- Edge case handling: Verified with improper rational functions, zero-pole cancellations, and high-degree denominators.
Transparency note: All calculations run entirely client-side in your browser. No input data is collected, stored, or transmitted. Results are for educational purposes; always verify critical engineering calculations independently.