A periodic function of period \( \frac{2\pi}{\omega} \) is defined by: \[ f(t) = \begin{cases} E \sin(\omega t), & 0 \leq t < \frac{\pi}{\omega} \\ 0, & \frac{\pi}{\omega} \leq t < \frac{2\pi}{\omega} \end{cases} \] where \( E \) and \( \omega \) are positive constants. Show that the Laplace transform of \( f(t) \) is given by: \[ L\{ f(t) \} = \frac{E \omega}{(s^2 + \omega^2)} (1 - e^{-\frac{\pi s}{\omega}}) \]