LaTeX Laplace transform

\[ \mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st} f(t) \, dt \] For example, the Laplace transform of \( f(t) = e^{at} \) is: \[ \mathcal{L}\{e^{at}\} = \int_0^{\infty} e^{-st} e^{at} \, dt = \int_0^{\infty} e^{-(s-a)t} \, dt = \frac{1}{s-a}, \text{for } s > a \]