Let $\tau_a=\inf\{t: B_t=a\}$, the hitting time of the standard Brownian motion to reach the boundary $a$.
This is easily derived
$E(e^{-\lambda \tau_a})=e^{-|a|\sqrt{2\lambda}}$
But I am having a problem of using this formula to get the moments of $\tau_a$ by matching the coefficients of terms for each power of $\lambda$, specifically, the LHS, if expanded, has all integer powers of $\lambda$; however, the RHS has the various terms of $\sqrt{\lambda}$. How to reconcile the difference here? Could somebody please help? Thanks.