Laplace Transform Calculator
What is the Laplace Transform?
The Laplace transform is a powerful mathematical tool that converts a function of time f(t) into a function of complex frequency F(s). It transforms differential equations into algebraic equations, making them easier to solve.
The Laplace transform of a function f(t) is defined as:
F(s) = ℒ{f(t)} = ∫0∞ e-stf(t)dt
Where s is a complex number frequency parameter s = σ + iω, with real numbers σ and ω.
Applications of Laplace Transforms
Laplace transforms are widely used in various fields of science and engineering:
- Solving differential equations in control systems
- Analyzing electrical circuits and systems
- Signal processing and communications
- Mechanical vibrations and structural analysis
- Heat transfer and diffusion problems
Common Laplace Transform Properties
| Property | Time Domain f(t) | Laplace Domain F(s) |
|---|---|---|
| Linearity | af(t) + bg(t) | aF(s) + bG(s) |
| Time Shift | f(t-a)u(t-a) | e-asF(s) |
| Frequency Shift | eatf(t) | F(s-a) |
| Time Scaling | f(at) | (1/a)F(s/a) |
| Differentiation | df(t)/dt | sF(s) - f(0) |
| Integration | ∫0tf(τ)dτ | F(s)/s |
How to Use the Laplace Transform Calculator
- Enter your function in the time domain f(t)
- Click "Calculate Laplace Transform" to get the result
- View the step-by-step solution to understand the process
- Explore the examples to see common Laplace transforms
Inverse Laplace Transform
The inverse Laplace transform converts a function from the frequency domain back to the time domain. It is defined as:
f(t) = ℒ-1{F(s)} = (1/2πi)∫γ-i∞γ+i∞ estF(s)ds
In practice, inverse Laplace transforms are typically found using tables of known transform pairs and properties rather than direct integration.