Laplace Transform
Definition: F(s) = L{f(t)} = โซโ^โ f(t)ยทe^(-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/sยฒ | Re(s) > 0 |
| tโฟ | n!/sโฟโบยน | Re(s) > 0 |
| e^(at) | 1/(s-a) | Re(s) > a |
| tยทe^(at) | 1/(s-a)ยฒ | Re(s) > a |
| tโฟยทe^(at) | n!/(s-a)โฟโบยน | Re(s) > a |
| sin(ฯt) | ฯ/(sยฒ+ฯยฒ) | Re(s) > 0 |
| cos(ฯt) | s/(sยฒ+ฯยฒ) | Re(s) > 0 |
| sinh(at) | a/(sยฒ-aยฒ) | Re(s) > |a| |
| cosh(at) | s/(sยฒ-aยฒ) | Re(s) > |a| |
| e^(at)ยทsin(ฯt) | ฯ/((s-a)ยฒ+ฯยฒ) | Re(s) > a |
| e^(at)ยทcos(ฯt) | (s-a)/((s-a)ยฒ+ฯยฒ) | Re(s) > a |
| ฮด(t) (Dirac delta) | 1 | All s |
| u(t) (unit step) | 1/s | Re(s) > 0 |
| tยทsin(ฯt) | 2ฯs/(sยฒ+ฯยฒ)ยฒ | Re(s) > 0 |
| tยทcos(ฯt) | (sยฒ-ฯยฒ)/(sยฒ+ฯยฒ)ยฒ | Re(s) > 0 |
Properties
| Property | Time Domain | s-Domain |
|---|---|---|
| Linearity | af(t) + bg(t) | aF(s) + bG(s) |
| Time Shift | f(t-a)ยทu(t-a) | e^(-as)ยทF(s) |
| Frequency Shift | e^(at)ยทf(t) | F(s-a) |
| Scaling | f(at) | (1/a)ยทF(s/a) |
| 1st Derivative | f'(t) | sF(s) - f(0) |
| 2nd Derivative | f''(t) | sยฒF(s) - sf(0) - f'(0) |
| Integration | โซโแต f(ฯ) dฯ | F(s)/s |
| Multiplication by t | tยทf(t) | -F'(s) |
| Convolution | f(t) * g(t) | F(s)ยทG(s) |
| Initial Value Theorem | f(0โบ) | lim(sโโ) sF(s) |
| Final Value Theorem | f(โ) | lim(sโ0) sF(s) |