Compute f(t) from F(s) using robust numerical Stehfest algorithm & symbolic matching. Now supports mathematical constants (π, e) in time range inputs.
Definition: ℒ⁻¹{F(s)} = f(t) such that ℒ{f(t)} = F(s). For a given F(s), the time-domain function f(t) is uniquely determined (for t ≥ 0).
The inverse Laplace transform is a mathematical operation that converts a function F(s) from the complex frequency domain back to the time domain f(t). It is fundamental in solving linear ordinary differential equations, analyzing control systems, and studying electrical circuits.
Formal Definition (Bromwich integral):
f(t) = ℒ-1{F(s)} = \frac{1}{2πi} ∫_{γ-i∞}^{γ+i∞} F(s) e^{st} ds
where γ is a real number chosen so that the integration path lies to the right of all singularities of F(s). For most engineering applications, we rely on transform tables and numerical methods.
For rational functions (ratios of polynomials), the inverse can be found by expanding F(s) into simpler fractions and using known transform pairs. For example:
| Property | F(s) | f(t) = ℒ⁻¹{F(s)} |
|---|---|---|
| Linearity | αF(s) + βG(s) | αf(t) + βg(t) |
| First shift (s-domain) | F(s+a) | e^{-at}f(t) |
| Second shift (t-domain) | e^{-as}F(s) | f(t-a) u(t-a) |
| Scaling | F(s/a)/a | f(at) |
| Derivative in s | F'(s) | -t f(t) |
| Integral in s | ∫_s^∞ F(σ) dσ | f(t)/t |
| Convolution | F(s)G(s) | ∫₀ᵗ f(τ)g(t-τ)dτ |
| Initial value | lim_{s→∞} sF(s) | f(0⁺) |
| Final value | lim_{s→0} sF(s) | lim_{t→∞} f(t) (if limit exists) |
| F(s) | f(t), t ≥ 0 | Application |
|---|---|---|
| 1 | δ(t) (Dirac impulse) | Impulse response |
| 1/s | 1 (unit step u(t)) | Step response |
| 1/s² | t | Ramp |
| 1/(s+a) | e^{-at} | Exponential decay |
| 1/(s+a)² | t e^{-at} | Damped ramp |
| ω/(s²+ω²) | sin(ωt) | Undamped oscillation |
| s/(s²+ω²) | cos(ωt) | Undamped oscillation |
| 1/[(s+a)²+ω²] | (1/ω) e^{-at} sin(ωt) | Damped sine |
| (s+a)/[(s+a)²+ω²] | e^{-at} cos(ωt) | Damped cosine |
| e^{-as}/s | u(t-a) (delayed step) | Time delay |
| 1/(s(s+a)) | (1/a)(1 - e^{-at}) | First-order system step response |
| ωₙ²/(s²+2ζωₙs+ωₙ²) | Step response of 2nd-order system | Control theory |
The inverse Laplace transform exists and is unique for functions F(s) that are analytic in the half-plane Re(s) > α and satisfy |F(s)| → 0 as |s| → ∞. In practice, any physically realizable system meets these conditions.
The Stehfest method approximates f(t) by a weighted sum of F(s) evaluated at specific points:
f(t) ≈ \frac{\ln 2}{t} \sum_{i=1}^{N} V_i \, F\!\left(\frac{i \ln 2}{t}\right)
where N is an even integer and V_i are precomputed coefficients. This algorithm is fast and works well for smooth, non-oscillatory functions. Higher N improves accuracy but may cause numerical instability. Our calculator uses N=10 as default, providing a good balance.
Applications of Inverse Laplace Transform: