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.
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
A note on where this came from. Pierre-Simon Laplace introduced the transform in the late 18th century, but it sat as a mathematical curiosity for over a hundred years. Oliver Heaviside revived it in the 1880s for telegraph circuit analysis, and by the time control theory and electrical engineering matured in the early 20th century, the transform had become the working tool it is today. Most students meet it first in a differential equations course and again, two years later, in a circuits or controls course. The second time is usually when it clicks.
Why this works at all. The trick is in the derivative property: the Laplace transform turns d/dt into multiplication by s. A differential equation with multiple derivatives becomes an ordinary polynomial equation, you solve it algebraically in the s-domain, and then invert. The initial conditions ride along inside the algebra rather than needing a separate substitution step, which is why this method is so much faster than the classical method-of-undetermined-coefficients approach for linear ODEs with non-trivial forcing.