Laplace Transform Calculator
Forward & inverse L{f(t)} · partial fractions · ROC · step-by-step
Quick examples
Common Laplace pairs
| f(t) | F(s) | ROC |
|---|---|---|
| Re(s) > 0 | ||
| Re(s) > 0 | ||
| Re(s) > −a | ||
| Re(s) > 0 | ||
| Re(s) > 0 | ||
| Re(s) > −a | ||
| Re(s) > −a | ||
| all s | ||
| Re(s) > 0 |
Syntax help
sin(2*t) cos(5*t)
e^(-3*t) exp(-t)
sinh(t) cosh(t)
Heaviside(t-2) DiracDelta(t)
1/(s^2+1) s/(s^2+9)
Multiply:
t*e^(-t) not te^(-t)
Powers: t^2 Constants: pi, e
Enter an expression and click Compute
Compute forward or inverse Laplace transforms with step-by-step solutions.
Compute a transform to see the time-domain plot.
What is the Laplace Transform?
The Laplace transform is an integral transform that converts a function of time f(t) into a function of complex frequency F(s). It is defined as L{f(t)} = ∫0∞ f(t) e−st dt, where s = σ + jω is a complex variable.
The transform is essential in engineering and physics for solving ordinary differential equations, analyzing linear time-invariant (LTI) systems, and designing control systems and electrical circuits.
Key Properties
Linearity
L{af(t) + bg(t)} = aF(s) + bG(s). The transform distributes over addition and scalar multiplication.
First Shifting Theorem
L{e−atf(t)} = F(s+a). Multiplication by an exponential shifts the s-domain by a.
Derivative Property
L{f'(t)} = sF(s) − f(0). Differentiation becomes multiplication by s, converting ODEs to algebra.
Convolution Theorem
L{f*g} = F(s)·G(s). Convolution in the time domain becomes multiplication in the s-domain.
Applications
Circuit Analysis
Analyze RLC circuits by transforming differential equations into algebraic equations in the s-domain.
Control Systems
Transfer functions H(s) describe system input-output relationships. Used for stability and feedback analysis.
Signal Processing
Filter design and frequency response analysis of continuous-time linear systems.
Differential Equations
Solve initial-value problems by converting ODEs to algebraic equations, solving for F(s), then inverting.