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Let $S_n$ be a random walk drifting to -∞. We obtain an asymptotic expansion for the distribution of the supremum of $S_n$ which takes into account the influence of the roots of the equation $1-{\int }_{ℝ}{e}^{sx}F\left(dx\right)=0,F$ being the underlying distribution. An estimate, of considerable generality, is given for the remainder term by means of submultiplicative weight functions. A similar problem for the stationary distribution of an oscillating random walk is also considered. The proofs rely on two general theorems for Laplace transforms....