Let \(\tau=\tau_{(a, b)^{c}}^{\circ}\) be the first exit time of a Brownian motion from the interval \((a, b)\).

a) Find \(\mathbb{E}^{x} e^{-\lambda \tau}\) for all \(x \in(a, b)\) and \(\lambda>0\).

b) Find \(\mathbb{E}^{x}\left(e^{-\lambda \tau} \mathbb{1}_{\left\{B_{\tau}=aight\}}ight)\) for all \(x \in(a, b)\) and \(\lambda>0\).