Laplace Transform Calculator
Laplace Transform Calculator 📊
Find the Laplace transform of elementary functions using our online calculator. Uses standard transform tables for quick and accurate calculations.
🔍 What is Laplace Transform?
The Laplace transform is an integral transform that converts a function of time f(t) into a function of complex variable F(s). It's widely used for solving differential equations and analyzing dynamic systems.
📐 Basic Formula
L{f(t)} = F(s) = ∫[0→∞] f(t)·e^(-st) dt
⚡ Supported Functions:
- Constant: L{c} = c/s
- Power: L{t^n} = n!/s^(n+1)
- Exponential: L{e^(at)} = 1/(s-a)
- Sine: L{sin(at)} = a/(s²+a²)
- Cosine: L{cos(at)} = s/(s²+a²)
- Hyperbolic functions
- Products with t
🎯 Applications:
Laplace transforms are essential in electrical engineering, control systems, signal processing, and solving linear differential equations with constant coefficients.
Frequently Asked Questions
What is the Laplace transform?
It's an integral transform that converts a time-domain function into a complex frequency-domain function. It simplifies solving differential equations by converting them to algebraic equations.
Why use Laplace transforms?
Main applications include solving linear differential equations, analyzing electrical circuits, control systems design, and signal processing in engineering.
How to find the original function from its transform?
Use the inverse Laplace transform or lookup tables that match transforms to their original functions.
What does parameter 's' represent?
s is a complex variable s = σ + jω, where σ is the real part and ω is the imaginary part (frequency).
What's the region of convergence?
It's the set of values of s for which the Laplace transform integral converges. For most elementary functions, it's Re(s) > some real number.