Laplace Transform Calculator
Calculate Laplace transforms and inverse transforms for mathematical functions. Comprehensive transform table, convergence analysis, and step-by-step solutions for differential equations.
Function Library
Power Functions
Exponential Functions
Trigonometric Functions
Special Functions
Function Input
Use standard notation: t^2, e^(at), sin(at), cos(at), etc.
Syntax Examples:
• Powers: t, t^2, t^3
• Exponentials: e^t, e^(-t), e^(2t)
• Trigonometric: sin(t), cos(2t), sin(3t)
• Special: δ(t), u(t-a)
Common Transform Pairs
| f(t) | F(s) | Domain | Description |
|---|---|---|---|
| 1 | 1/s | s > 0 | Unit step function |
| t | 1/s² | s > 0 | Linear ramp |
| t^n | n!/s^(n+1) | s > 0, n > -1 | Power function |
| t² | 2/s³ | s > 0 | Quadratic function |
| t³ | 6/s⁴ | s > 0 | Cubic function |
| e^(at) | 1/(s-a) | s > a | Exponential function |
| te^(at) | 1/(s-a)² | s > a | Exponential with linear factor |
| t^n e^(at) | n!/(s-a)^(n+1) | s > a | Power times exponential |
Showing 8 of 20 transform pairs. Full table available in generated report.
Laplace Transform
Definition:
L{f(t)} = F(s) = ∫₀^∞ f(t)e^(-st) dt
Convergence:
Transform exists when the integral converges, typically for s with sufficiently large real part.
Linearity:
L{af(t) + bg(t)} = aF(s) + bG(s)
Inverse Transform:
f(t) = L⁻¹{F(s)} recovers original function from its transform.
Applications
Differential Equations:
Convert differential equations to algebraic equations in s-domain.
Control Systems:
Analyze system stability, transfer functions, and frequency response.
Signal Processing:
Filter design, system analysis, and signal transformation.
Circuit Analysis:
Solve RLC circuits, analyze transient and steady-state responses.