The Laplace Calculator helps to compute the Laplace transform of a given function. The Laplace transform is widely used in engineering and physics to solve differential equations and to transform functions from the time domain to the s-domain.
Inputs
- Function (f(t)): The function of time ( t ) to be transformed.
- Variable (s): The complex frequency domain variable used in the transformation.
Formulas and Calculations
The Laplace transform of a function ( f(t) ) is given by the formula:
[ \mathcal{L}{f(t)} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt ]
where:
- ( \mathcal{L}{f(t)} ) = Laplace transform of ( f(t) )
- ( F(s) ) = Resulting function in the s-domain
- ( s ) = Complex frequency domain variable
- ( t ) = Time variable
Example Calculation
Input:
- Function ( f(t) ): ( e^{at} )
- Apply the Laplace Transform Formula:
[ \mathcal{L}{e^{at}} = \int_{0}^{\infty} e^{-st} e^{at} \, dt ]
- Combine the Exponents:
[ \mathcal{L}{e^{at}} = \int_{0}^{\infty} e^{-(s-a)t} \, dt ]
- Integrate with Respect to ( t ):
[ \mathcal{L}{e^{at}} = \left[ \frac{e^{-(s-a)t}}{-(s-a)} \right]_{0}^{\infty} ]
- Evaluate the Limits:
[ \mathcal{L}{e^{at}} = \left( \frac{0}{-(s-a)} – \frac{1}{-(s-a)} \right) ]
[ \mathcal{L}{e^{at}} = \frac{1}{s-a} ]
Result:
The Laplace transform of ( e^{at} ) is ( \frac{1}{s-a} ).
Summary
For a function ( e^{at} ), the Laplace transform is ( \frac{1}{s-a} ). The Laplace Calculator simplifies this transformation, enabling users to solve complex differential equations efficiently. By providing the function ( f(t) ) and the variable ( s ), the calculator returns the Laplace transform, making it an essential tool in engineering and physics.
Conclusion
The Laplace Calculator is a valuable tool for transforming time-domain functions into the s-domain, facilitating the solution of differential equations and analysis of dynamic systems. By inputting the function ( f(t) ) and the variable ( s ), users can quickly determine the Laplace transform.