The asymptotic behavior of the solutions of an elliptic Dirichlet linear problem, where the coefficients vary periodically has been considered by several authors. However, the classical approximation (corrector) of the solutions obtained by homogenization is not good near the boundary, where a boundary layer appears. Here, we show how this problem can be deal by an adaptation of the two-scale convergence method for quasi-periodic functions. We obtain a boundary layer term, which improves the approximation given by the classical corrector.