Inverse Laplace Solver
Convert s-domain functions $F(s)$ back to time-domain $f(t)$.
Handles rational functions, shifts, and standard table lookups.
Input Function F(s)
Enter function of ‘s’ (e.g., 1/(s+2), s/(s^2+4))
Inverse Definition
The Inverse Laplace Transform recovers the time-domain signal from its s-domain representation. It usually relies on Partial Fraction Decomposition to break complex expressions into standard forms found in tables.
Result f(t)
Time Domain f(t)
Method of Partial Fractions & Inverses
Calculating the Inverse Laplace Transform often requires algebraic manipulation before looking up the answer in a table. The most common technique is Partial Fraction Expansion.
Standard Inverse Pairs
- Decay: $\mathcal{L}^{-1}\left\{\frac{1}{s+a}\right\} = e^{-at}$
- Oscillation: $\mathcal{L}^{-1}\left\{\frac{\omega}{s^2 + \omega^2}\right\} = \sin(\omega t)$ and $\mathcal{L}^{-1}\left\{\frac{s}{s^2 + \omega^2}\right\} = \cos(\omega t)$
- Powers: $\mathcal{L}^{-1}\left\{\frac{n!}{s^{n+1}}\right\} = t^n$
- Damped Oscillation (First Shift): $$ \mathcal{L}^{-1}\left\{\frac{\omega}{(s-a)^2 + \omega^2}\right\} = e^{at}\sin(\omega t) $$
Strategies for Students
1. Factor the Denominator: Determine if the poles are real distinct, repeated, or complex conjugates.
2. Complete the Square: If the denominator is an irreducible quadratic (e.g., $s^2+4s+13$), rewrite it as $(s+2)^2 + 3^2$ to use the shift theorems.
3. Linearity: Break the numerator apart. For example, $\frac{s+1}{s^2+1} = \frac{s}{s^2+1} + \frac{1}{s^2+1} \rightarrow \cos(t) + \sin(t)$.