Laplace Transform Examples

Master Laplace transforms with these step-by-step examples covering the most common functions

Showing 6 examples

BeginnerTrigonometric

Basic Sine Function

Input: f(t) =
sin(3*t)
Output: F(s) =
3/(s² + 9)
Convergence Region:
Re(s) > 0

Learn how to transform basic sine functions

Steps:
Apply the Laplace transform definition
Use the sine transform formula: L{sin(at)} = a/(s² + a²)
+ 1 more steps
BeginnerExponential

Exponential Decay

Input: f(t) =
exp(-2*t)
Output: F(s) =
1/(s + 2)
Convergence Region:
Re(s) > -2

Understanding exponential decay transforms

Steps:
Apply the exponential transform formula
L{e^(at)} = 1/(s - a)
+ 1 more steps
IntermediateEngineering

Damped Oscillation

Input: f(t) =
exp(-2*t)*cos(3*t)
Output: F(s) =
(s + 2)/((s + 2)² + 9)
Convergence Region:
Re(s) > -2

Common in control systems and circuit analysis

Steps:
Use the frequency shifting property
L{e^(at)f(t)} = F(s-a)
+ 1 more steps
IntermediateStep Functions

Unit Step Function

Input: f(t) =
heaviside(t-2)
Output: F(s) =
exp(-2*s)/s
Convergence Region:
Re(s) > 0

Essential for control systems and signal processing

Steps:
Apply the time shifting property
L{u(t-a)} = e^(-as)/s
+ 1 more steps
BeginnerPolynomial

Polynomial Function

Input: f(t) =
t^2
Output: F(s) =
2/s³
Convergence Region:
Re(s) > 0

Basic polynomial transforms

Steps:
Use the power rule: L{tⁿ} = n!/s^(n+1)
For n = 2, we get 2!/s³
+ 1 more steps
AdvancedImpulse

Impulse Function

Input: f(t) =
dirac(t)
Output: F(s) =
1
Convergence Region:
All s

Delta function - fundamental in system analysis

Steps:
Apply the definition of Dirac delta
L{δ(t)} = ∫₀^∞ δ(t)e^(-st) dt
+ 1 more steps

Ready to try these transforms?

Use our calculator to verify these examples and solve your own problems