Abstract. "Let $\{S_n:n\ge 0\}$ be a random walk having normally distributed increments with mean $\theta$ and variance 1, and let $\tau$ be the time at which the random walk first takes a positive value, so that $S_\tau$ is the first ladder height. Then the expected value $E_\theta S_\tau$, originally defined for  $\theta>0$, may be extended to be an analytic function of the complex variable $\theta$ throughout the complex plane, with the exception of certain branch point  singularities.  In particular, the coefficients in a Taylor expansion about $\theta=0$ may be written explicitly as simple expressions involving the Riemann zeta function. Previously only the first coefficient of the series developed here was known; this term has been  used extensively in developing approximations for boundary crossing problems for Gaussian random walks.   Knowledge of the complete series makes more refined results possible; we apply it to derive asymptotics for boundary crossing probabilities and the limiting expected overshoot."