478.Laplace Transform of an Exit Time

Let $W_t$ be a standard Brownian motion and let $T_a = \inf\{t > 0 : |W_t| > a\}$ be the first exit time from $(-a, a)$. The Laplace transform is $E[e^{-\lambda T_a}] = \mathrm{sech}(a\sqrt{2\lambda})$. Evaluate it at $\lambda = 4$ and $a = \ln 2$.