Let $\{S_n: n \geq 0\}$ 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 positive $\theta$, maybe extended to be an analytic function of the
complex variable $\theta$ throughout the entire complex plane, with the
exception of certain branch point sin-gularities. 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.