Laplace Transform Calculator
Compute the Laplace transform of any function with step-by-step solutions. This powerful tool helps solve differential equations and analyze linear systems in engineering and physics.
Laplace Transform Results
Step-by-Step Solution
Select a function and click "Calculate" to see the step-by-step solution.
Common Laplace Transforms
| Function \( f(t) \) | Laplace Transform \( F(s) \) |
|---|---|
| \( 1 \) | \( \frac{1}{s} \) |
| \( t \) | \( \frac{1}{s^2} \) |
| \( t^n \) (n integer) | \( \frac{n!}{s^{n+1}} \) |
| \( e^{at} \) | \( \frac{1}{s-a} \) |
| \( \sin(at) \) | \( \frac{a}{s^2 + a^2} \) |
| \( \cos(at) \) | \( \frac{s}{s^2 + a^2} \) |
About the Laplace Transform Calculator
The Laplace transform is an integral transform used to convert a function of a real variable (usually time) to a function of a complex variable (frequency). It's widely used in engineering, physics, and applied mathematics to solve differential equations and analyze linear systems.
How Laplace Transforms Work
The Laplace transform of a function \( f(t) \) is defined as:
\[ \mathcal{L}\{f(t)\} = F(s) = \int_0^\infty e^{-st} f(t) \, dt \]
where \( s \) is a complex number frequency parameter.
Key Properties of Laplace Transforms
Laplace transforms have several important properties that make them useful for solving differential equations:
- Linearity: \( \mathcal{L}\{af(t) + bg(t)\} = aF(s) + bG(s) \)
- Differentiation: \( \mathcal{L}\{f'(t)\} = sF(s) - f(0) \)
- Integration: \( \mathcal{L}\left\{\int_0^t f(\tau) d\tau\right\} = \frac{F(s)}{s} \)
- Time shifting: \( \mathcal{L}\{f(t-a)u(t-a)\} = e^{-as}F(s) \)
- Frequency shifting: \( \mathcal{L}\{e^{at}f(t)\} = F(s-a) \)
Applications of Laplace Transforms
Laplace transforms are used in various fields:
- Electrical Engineering: Analyzing circuits and control systems
- Mechanical Engineering: Solving vibration and heat transfer problems
- Physics: Solving wave equations and other PDEs
- Economics: Modeling dynamic systems
- Signal Processing: Analyzing linear time-invariant systems
Inverse Laplace Transform
The inverse Laplace transform converts a function \( F(s) \) back to its original form \( f(t) \):
\[ \mathcal{L}^{-1}\{F(s)\} = f(t) = \frac{1}{2\pi i} \lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT} e^{st} F(s)\, ds \]
In practice, we often use tables of known transforms and partial fraction decomposition to find inverse transforms.
Using This Calculator
This calculator can:
- Compute Laplace transforms of common functions
- Calculate inverse Laplace transforms
- Show step-by-step solutions
- Visualize functions and their transforms
- Provide reference tables of common transforms