Laplace Transform Calculator
Look up Laplace transforms for common functions.
Select function type, enter parameters, and get both f(t) and F(s) in formatted form.
Laplace Transform F(s)
The Laplace Transform
The Laplace transform converts a function of time f(t) into a function of complex frequency F(s). It is defined as:
F(s) = ∫₀^∞ f(t) · e^(−st) dt
This converts differential equations in t into algebraic equations in s — making them much easier to solve.
Key Properties
| Property | Time Domain | s-Domain |
|---|---|---|
| Linearity | af(t) + bg(t) | aF(s) + bG(s) |
| Derivative | f’(t) | sF(s) − f(0) |
| 2nd derivative | f’’(t) | s²F(s) − sf(0) − f’(0) |
| Time shift | f(t−a)u(t−a) | e^(−as)F(s) |
| Frequency shift | e^(at)f(t) | F(s−a) |
| Convolution | (f*g)(t) | F(s)·G(s) |
Common Transform Pairs
| f(t) | F(s) | Condition |
|---|---|---|
| 1 (constant) | 1/s | s > 0 |
| t^n | n!/s^(n+1) | s > 0 |
| e^(at) | 1/(s−a) | s > a |
| sin(ωt) | ω/(s²+ω²) | s > 0 |
| cos(ωt) | s/(s²+ω²) | s > 0 |
| e^(at)sin(ωt) | ω/((s−a)²+ω²) | s > a |
| e^(at)cos(ωt) | (s−a)/((s−a)²+ω²) | s > a |
| u(t−a) | e^(−as)/s | s > 0 |
| δ(t−a) | e^(−as) | all s |
Applications
- Control systems: analyzing stability of feedback loops
- Circuit analysis: solving RLC circuits without differential equations
- Signal processing: designing filters and analyzing frequency response
- Mechanical vibrations: modeling spring-mass-damper systems