Laplace Transform
Definition: F(s) = L{f(t)} = ∫₀^∞ f(t)·e^(-st) dt (Re(s) > convergence abscissa)
Common Transform Pairs
| f(t) | F(s) = L{f(t)} | Region of Convergence |
|---|---|---|
| 1 | 1/s | Re(s) > 0 |
| t | 1/s² | Re(s) > 0 |
| tⁿ | n!/sⁿ⁺¹ | Re(s) > 0 |
| e^(at) | 1/(s-a) | Re(s) > a |
| t·e^(at) | 1/(s-a)² | Re(s) > a |
| tⁿ·e^(at) | n!/(s-a)ⁿ⁺¹ | Re(s) > a |
| sin(ωt) | ω/(s²+ω²) | Re(s) > 0 |
| cos(ωt) | s/(s²+ω²) | Re(s) > 0 |
| sinh(at) | a/(s²-a²) | Re(s) > |a| |
| cosh(at) | s/(s²-a²) | Re(s) > |a| |
| e^(at)·sin(ωt) | ω/((s-a)²+ω²) | Re(s) > a |
| e^(at)·cos(ωt) | (s-a)/((s-a)²+ω²) | Re(s) > a |
| δ(t) (Dirac delta) | 1 | All s |
| u(t) (unit step) | 1/s | Re(s) > 0 |
| t·sin(ωt) | 2ωs/(s²+ω²)² | Re(s) > 0 |
| t·cos(ωt) | (s²-ω²)/(s²+ω²)² | Re(s) > 0 |
Properties
| Property | Time Domain | s-Domain |
|---|---|---|
| Linearity | af(t) + bg(t) | aF(s) + bG(s) |
| Time Shift | f(t-a)·u(t-a) | e^(-as)·F(s) |
| Frequency Shift | e^(at)·f(t) | F(s-a) |
| Scaling | f(at) | (1/a)·F(s/a) |
| 1st Derivative | f'(t) | sF(s) - f(0) |
| 2nd Derivative | f''(t) | s²F(s) - sf(0) - f'(0) |
| Integration | ∫₀ᵗ f(τ) dτ | F(s)/s |
| Multiplication by t | t·f(t) | -F'(s) |
| Convolution | f(t) * g(t) | F(s)·G(s) |
| Initial Value Theorem | f(0⁺) | lim(s→∞) sF(s) |
| Final Value Theorem | f(∞) | lim(s→0) sF(s) |