Laplace Transform Calculator

What is Laplace Transform Calculator?

A Laplace Transform Calculator is a mathematical tool that converts functions from the time domain into the complex frequency domain. The Laplace transform is widely used in engineering, physics, and control systems to solve differential equations and analyze systems. By transforming a time-dependent function \(f(t)\) into a function of a complex variable \(s\), the Laplace transform simplifies calculations and allows easier manipulation.

The Laplace transform is defined as: \[ \mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st} f(t) \, dt \]

About the Laplace Transform Calculator

This calculator automates the Laplace transform process for a wide variety of functions, including polynomials, exponentials, sines, cosines, and combinations. Users can enter the time-domain function, and the calculator applies integration, linearity, and standard Laplace transform properties to provide a step-by-step solution. It can also handle inverse Laplace transforms to convert back to the time domain. Common formulas include:

\(\mathcal{L}\{1\} = \frac{1}{s}, \quad \mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}, \quad \mathcal{L}\{e^{at}\} = \frac{1}{s-a}, \quad \mathcal{L}\{\sin(bt)\} = \frac{b}{s^2+b^2}, \quad \mathcal{L}\{\cos(bt)\} = \frac{s}{s^2+b^2}\)

How to Use this Laplace Transform Calculator

1. Enter the time-domain function \(f(t)\) into the input field.
2. Click the Calculate button to compute the Laplace transform \(F(s)\).
3. The calculator provides a step-by-step breakdown, applying linearity, integration, and standard Laplace transform rules.
4. For inverse transforms, input the \(F(s)\) function and select Inverse Laplace to retrieve \(f(t)\).
5. Use results to solve differential equations, analyze systems, or study signal behavior efficiently.

Example:

For \(f(t) = e^{2t}\): \[ \mathcal{L}\{e^{2t}\} = \int_0^{\infty} e^{-st} e^{2t} \, dt = \int_0^{\infty} e^{-(s-2)t} dt = \frac{1}{s-2}, \quad \text{Re}(s) > 2 \]

This calculator makes Laplace transforms simple, accurate, and easy to understand for students, engineers, and researchers working with differential equations.