Inverse Laplace Transform Calculator
Convert functions from the s-domain back to the time domain. Essential for solving differential equations.
Common Inverse Laplace Transforms
| F(s) | f(t) = ℒ⁻¹{F(s)} | Name |
|---|---|---|
| a/s | a | Constant |
| 1/(s-a) | e^(at) | Exponential |
| n!/s^(n+1) | t^n | Power |
| a/(s²+a²) | sin(at) | Sine |
| s/(s²+a²) | cos(at) | Cosine |
| b/((s-a)²+b²) | e^(at)sin(bt) | Exp×Sine |
| (s-a)/((s-a)²+b²) | e^(at)cos(bt) | Exp×Cosine |
| a/(s²-a²) | sinh(at) | Sinh |
| s/(s²-a²) | cosh(at) | Cosh |
Key Laplace Transform Theorems
First Shifting Theorem
ℒ⁻¹{F(s-a)} = e^(at) × f(t)
Shifts in s-domain multiply by exponential in t-domain
Second Shifting Theorem
ℒ⁻¹{e^(-as)F(s)} = u(t-a)f(t-a)
Exponential in s-domain shifts function in t-domain
Linearity
ℒ⁻¹{aF + bG} = af(t) + bg(t)
Split complex expressions into simpler parts
Convolution
ℒ⁻¹{F(s)G(s)} = (f * g)(t)
Product becomes convolution integral
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Educational Note: This calculator covers standard Laplace transform pairs. For complex expressions, use partial fraction decomposition and combine results. Always verify solutions in academic contexts.
About Inverse Laplace Transform Calculator - F(s) to f(t)
Calculate inverse Laplace transforms with step-by-step solutions. Convert from F(s) in frequency domain back to f(t) in time domain.
Our **Inverse Laplace Transform Calculator** converts functions from the frequency domain F(s) back to the time domain f(t). Select from common transform types and get instant results with detailed explanations. Essential for differential equations and control systems.
The inverse Laplace transform is used extensively in engineering to solve differential equations. Given F(s), we find f(t) such that L{f(t)} = F(s). Our calculator handles exponentials, sinusoids, polynomials, and their combinations.
Learn the connection between s-domain and t-domain representations. Understand partial fractions, convolution, and transform tables through our step-by-step solutions. Perfect for electrical engineering and control theory students.
Key Features
Why Use This Tool?
Common Use Cases
Differential Equations: Solve linear ODEs using Laplace.
Control Systems: Analyze system response in time domain.
Circuit Analysis: Find transient responses.
Signal Processing: Inverse transform of transfer functions.
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How to Use
Select the transform type
Enter parameters (frequency, coefficients)
Click Calculate
View f(t) result and steps