Complete reference table of frequently used Laplace transforms
The Laplace transform is defined as: L{f(t)} = F(s) = ∫[0 to ∞] e^(-st) f(t) dt
| f(t) | F(s) | Region of Convergence |
|---|---|---|
| δ(t) | 1 | All s |
| u(t) | 1/s | Re(s) > 0 |
| t | 1/s² | Re(s) > 0 |
| t^n | n!/s^(n+1) | Re(s) > 0 |
| e^(at) | 1/(s-a) | Re(s) > a |
| te^(at) | 1/(s-a)² | Re(s) > a |
| f(t) | F(s) | Region of Convergence |
|---|---|---|
| sin(ωt) | ω/(s² + ω²) | Re(s) > 0 |
| cos(ωt) | s/(s² + ω²) | Re(s) > 0 |
| 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 |
| t·sin(ωt) | 2ωs/(s² + ω²)² | Re(s) > 0 |
| t·cos(ωt) | (s² - ω²)/(s² + ω²)² | Re(s) > 0 |
| f(t) | F(s) | Region of Convergence |
|---|---|---|
| sinh(at) | a/(s² - a²) | Re(s) > |a| |
| cosh(at) | s/(s² - a²) | Re(s) > |a| |
| t·sinh(at) | 2as/(s² - a²)² | Re(s) > |a| |
| t·cosh(at) | (s² + a²)/(s² - a²)² | Re(s) > |a| |
L{af(t) + bg(t)} = aF(s) + bG(s)
L{f(t-a)u(t-a)} = e^(-as)F(s)
L{e^(at)f(t)} = F(s-a)
L{f(at)} = (1/a)F(s/a)
L{f'(t)} = sF(s) - f(0)
L{∫[0 to t] f(τ)dτ} = F(s)/s