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.